Problem
COM-B2-M09-P001 Coefficient and Subset Choice
#1
★★☆☆☆ Level 2 of 5
Find the coefficient of \(x^4\) in \((1+x)^9\), and explain what combinatorial quantity it counts.
To obtain \(x^4\), choose \(x\) from four factors.
In the product \((1+x)^9\), each of the nine factors gives a choice: take \(1\) or take \(x\). Degree \(4\) occurs exactly when \(x\) is chosen from four factors. Therefore the coefficient is \(\binom{9}{4}=126\). It counts the four-element subsets of a nine-element set.
A basic check of the coefficient meaning in a binomial generating function.