https://artofproblemsolving.com/wiki/index.php?title=2008_Mock_ARML_2_Problems/Problem_3&feed=atom&action=history 2008 Mock ARML 2 Problems/Problem 3 - Revision history 2022-05-22T10:56:21Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2008_Mock_ARML_2_Problems/Problem_3&diff=26197&oldid=prev Azjps: solution 2008-05-30T00:08:33Z <p>solution</p> <p><b>New page</b></p><div>== Problem ==<br /> A variation of [[Pascal's triangle]] is constructed by writing the numbers &lt;math&gt;2&lt;/math&gt; and &lt;math&gt;3&lt;/math&gt; in the top row and writing each subsequent term as the sum of the two terms above it. Find the fifth term from the left in the thirteenth row.<br /> &lt;cmath&gt;<br /> \begin{tabular}{ccccccccccc} &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\<br /> &amp; &amp; &amp; &amp; 2 &amp; &amp; 3 &amp; &amp; &amp; &amp; \\<br /> &amp; &amp; &amp; 2 &amp; &amp; 5 &amp; &amp; 3 &amp; &amp; &amp; \\<br /> &amp; &amp; 2 &amp; &amp; 7 &amp; &amp; 8 &amp; &amp; 3 &amp; &amp; \\<br /> &amp; 2 &amp; &amp; 9 &amp; &amp; 15 &amp; &amp; 11 &amp; &amp; 3 &amp; \\<br /> 2 &amp; &amp; 11 &amp; &amp; 24 &amp; &amp; 26 &amp; &amp; 14 &amp; &amp; 3 \\<br /> \end{tabular}<br /> &lt;/cmath&gt;<br /> <br /> == Solution ==<br /> Note that this variation can be constructed using overlapping Pascal's triangles. Consider multiplying each of the terms in Pascal's triangles by &lt;math&gt;2&lt;/math&gt;; then taking another triangle and multiplying each of the terms by &lt;math&gt;3&lt;/math&gt;, shifting the resulting triangle one space to the right, and then summing the overlapping entries. Note that this preserves the desired recursion. <br /> <br /> It follows that the fifth term in the thirteenth row is &lt;math&gt;2{12 \choose 4} + 3{12 \choose 3} = \boxed{1650}&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{Mock ARML box|year = 2008|n = 2|num-b=2|num-a=4|source = 206880}}<br /> <br /> [[Category:Intermediate Combinatorics Problems]]</div> Azjps