# Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.5 The Binomial Theorem - 11.5 Assess Your Understanding - Page 857: 33

The coefficient of the term that contains $x^7$ is equal to $41, 472$.

#### Work Step by Step

According to the Binomial Theorem, the term containing $x^k$ in the expansion of $(p+q)^n$ can be determined as: $\displaystyle{n}\choose{n-k}$$p^{n-k} q^k$ Using the above formula and replacing $p$ with $2$ and $q$ with $3$, the term containing $x^7$ in the given expansion can be written as: $\dbinom{9}{9-7} \ (2)^{7}(3)^{9-7} =\dbinom {9} {2} (2)^{7}(3)^2 \\= \dfrac{9!}{2! \ 7!} (2)^7 (3)^2 \\=36 \times 128\times 9\\= 41, 472$ Therefore, the coefficient of the term that contains $x^7$ is equal to: $41, 472$

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