Write the indicated term of the binomial expansion.
-$( x + 2 y ) ^ { 11 ; } 9$ th term
A) $42,240 x ^ { 8 } y ^ { 3 }$
B) $42,240 x ^ { 3 } y ^ { 8 }$
C) $21,120 x ^ { 8 } y ^ { 3 }$
D) $21,120 \mathrm { x } ^ { 3 } \mathrm { y } ^ { 9 }$

Question

Quizplus · Accepted Answer

The $n$th term of the binomial expansion $(a + b)^n$ is given by $\binom{n}{k} a^{n-k}b^k$, where $\binom{n}{k}$ is the binomial coefficient. For the 9th term of the expansion $(x + 2y)^{11}$, $k = 9 - 1 = 8$ because the first term is when $k = 0$. Thus, we calculate the 9th term as $\binom{11}{8} x^{11-8} (2y)^8$. The binomial coefficient $\binom{11}{8} = \frac{11!}{8!(11-8)!} = 165$, and $(2y)^8 = 2^8y^8 = 256y^8$. Multiplying these together gives $165 \times 256 x^3 y^8 = 42,240 x^3 y^8$.

Write the Indicated Term of the Binomial Expansion (x+2y)11;9( x + 2 y ) ^ { 11 ; } 9(x+2y)11;9

Write the Indicated Term of the Binomial Expansion $( x + 2 y ) ^ { 11 ; } 9$