Problem
COM-B2-M09-P004 Sum of Chosen Numbers
#4
★★☆☆☆ Level 2 of 5
How many subsets of \(\{1,2,3,4,5,6,7\}\) have sum of elements equal to \(8\)?
Consider the product \(\prod_{i=1}^7(1+x^i)\).
We need the coefficient of \(x^8\) in \(\prod_{i=1}^7(1+x^i)\). The sum \(8\) is obtained as \(1+7\), \(2+6\), \(3+5\), \(1+2+5\), and \(1+3+4\). There are no other possibilities: for instance, if \(4\) is chosen with \(2\), the remaining \(2\) cannot be repeated; if \(6\) is chosen, only \(2\) remains, and so on. The answer is \(5\).
The problem trains the translation of weighted subsets into a product.