Find the coefficient $a$ of the term in the expansion of the binomial.
$\begin{array}{ll}
\text { Binomial } & \text { Term } \
(2 x-5 y)^{7} & a x^{2} y^{5}
\end{array}$

A) $a = 12$
B) $a = - 262,500$
C) $a = 14$
D) $a = - 78,125$
E) $\quad a = 2520$

Question

Find the coefficient $a$ of the term in the expansion of the binomial.
$\begin{array}{ll}
\text { Binomial } & \text { Term } \
(2 x-5 y)^{7} & a x^{2} y^{5}
\end{array}$

A) $a = 12$
B) $a = - 262,500$
C) $a = 14$
D) $a = - 78,125$
E) $\quad a = 2520$

Quizplus · Accepted Answer

The coefficient $a$ can be found using the binomial theorem, which in this case is given by $\binom{n}{k} a^{n-k} b^{k}$ for the term $a^{n-k} b^{k}$ in the expansion of $(a + b)^n$. Here, $n = 7$, $a = 2x$, $b = -5y$, and we want the term where $x^2y^5$, meaning $k = 5$. Thus, $\binom{7}{5} (2x)^{7-5} (-5y)^{5} = \binom{7}{5} (2x)^{2} (-5y)^{5}$. $\binom{7}{5} = 21$, so the coefficient $a$ is $21 \times 2^2 \times (-5)^5 = 21 \times 4 \times -3125 = -262,500$.

Find the Coefficient aaa Of the Term in the Expansion of the Binomial

Find the Coefficient $a$ Of the Term in the Expansion of the Binomial