Problem 57507. Easy Sequences 88: Double Summation of a Combinatorial Function

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Given two positive integers </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a\\ge b</w:t></w:r></w:customXml><w:r><w:t>, we define the function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(a,b)</w:t></w:r></w:customXml><w:r><w:t> as the sum of the reciprocals of the products of all combinations of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>1</w:t></w:r></w:customXml><w:r><w:t> to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> taken </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t> at a time. In Matlab we say that:    </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R = sum(1./prod(nchoosek(a,b)'))</w:t></w:r><w:r><w:t>. For example if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a=5</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b=3</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[  >> R = sum(1./prod(nchoosek(1:5,3)'))\n  R =\n      0.7083]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given a positive integer </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, we are asked to evaluate the following double summation:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>S(x) = \\sum_{n=3}^{x} \\sum_{k=2}^{n-1} R(n,k).</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=5</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>S(5) = \\sum_{n=3}^{5} \\sum_{k=2}^{n-1} R(n,k) = \\sum_{k=2}^2 R(3,k)+\\sum_{k=2}^3 R(4,k)+\\sum_{k=2}^4 R(5,k).</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In which, in Matlab, evaluates to:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[  >> S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\n  S =\n      5.5833]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Please round-off </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>S</w:t></w:r></w:customXml><w:r><w:t> to the nearest whole number. Therefore, the final output for the above example is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>S=6</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 435px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 757.578125px 217.5px; transform-origin: 757.578125px 217.5px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Given two positive integers </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">a</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">b</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, we define the function </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="46" height="19" style="width: 46px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> as the sum of the reciprocals of the products of all combinations of </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">1</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> to </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">a</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> taken </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">b</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> at a time. In Matlab we say that:    </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">R = sum(1./prod(nchoosek(a,b)'))</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. For example if </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 754.578125px 30px; transform-origin: 754.578125px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt; R = sum(1./prod(nchoosek(1:5,3)'))</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  R =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">      0.7083</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Given a positive integer </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">, we are asked to evaluate the following double summation:</span></span></div><div style="block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">                </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" width="143.5" height="46" style="width: 143.5px; height: 46px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">For example for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAASaADAAQAAAABAAAAJAAAAABLVRfgAAAC/ElEQVRoBe2XS4iOURjHPzIxLtkIjVtyWdgqshgWymVcFsKCKGSh2ZBYEFv3KSlZoayUWCgbl9XUsECUTNJ8SVjIEJrI9fcf3zOdjjPf+77zjrzzfeep33ee85znnO88z3vOec9bKkWJGYgZiBmIGRhSGRg2pGabf7JNDDElMIzysBBaYGWgva5MHUT7qwqtdZWNQLDNVZKjxH2EcYF+pREhY43a9hPXdzgOPwIx3sD2KWCvG9M8Iv0Jl+om4gEEep4+2lLrod5eVqnSpTfaV7AD+x36ddgCYyC3NDLCGjgMB2EsuLKcyglY6xoLph9jPpYgv+ymbVOe+WpwHWQ65Gzwk86Apxz7Z/TRTluR1L1M5h4oFovDLw8NdMJz6aiVMxWUBA38FCR7QDbt9S64CWnkKk5a7nlZmubPPB+dRdNhK5TBT9Q6bLnkGr1t0GXoX0BbLasomTZOnnJF1j/2/EdS15XAPas6qQ/3/HqraU96rZw2ZwBtu31OPa2qFTArrXMVPz20t1Xa0zbtwvGs49yM3u7UM6n6rrEn/wK9qOdPpqAqzped2DaHBggur4DjQ2zaYpL70NOr1cbPFSeMiY7ep6ZNkj5fdFBLFv0paub3gRPJB0fvU9N+u+kMmlDpNZlyNjyv1LMUerstydKhH9+N2G/305bVbDtE/d5k7Wz+q1F0V9IBZ+fSNmvMWBbl7eZOu6US13vKRrfB9KS32yQcH8NFOAq636jPBdgODXAGDkA3JEnR3m6a7y3QvE7DbkgUbb9VMAdGQTt0gG3LR+haTVqWsp2DO5CUbFz+i+iTowxPoBX8M3gnNsXTCcFDG/tfsgOLOgld43UXmQEmR1Cs/TW6Vpbbbn5FKXUG2nxV6oEvhvnQBrI9gyZILRvwtEG1ffw32UxsdkvVHtblq8iygMm9AovJLWXX9kq884W2iS6O40FbrQd80bKUz10YjFuvP/5g1xsYUA97GighL6EMXfANosQMxAzEDMQMxAzEDMQMxAzEDPy7DPwGCjXem+ac7dYAAAAASUVORK5CYII=" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" width="395.5" height="46" style="width: 395.5px; height: 46px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">In which, in Matlab, evaluates to:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 754.578125px 30px; transform-origin: 754.578125px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt; S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  S =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">      5.5833</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Please round-off </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">S</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> to the nearest whole number. Therefore, the final output for the above example is </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="37.5" height="18" style="width: 37.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. </span></span></div></div></div>

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Solution 10253910

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William on 10 Jan 2023

That took a bit of work!

Ramon Villamangca on 11 Jan 2023

Good work, though...

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Problem 57507. Easy Sequences 88: Double Summation of a Combinatorial Function

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Problem 57507. Easy Sequences 88: Double Summation of a Combinatorial Function

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