Question

For $(1+a-b+c{)}^9$, which of the following is/are correct?

• A. The coefficient of $a^3{b}^{4}c$ is $2520$
• B. The number of terms in the expansion is $220$
• C. The number of terms in the expansion is $110$
• D. The coefficient of $a^3{b}^{4}c$ is $1260$

Answer 1

Answer: (A), (B)

The correct options are
A The coefficient of $a^3{b}^{4}c$ is $2520$
B The number of terms in the expansion is $220$

Given expansion is $(1+a-b+c{)}^9$, so
The number of terms
$={}^{9+4-1}{C}_{4-1}={}^{12}{C}_3=\frac{12\cdot 11\cdot 10}{6}=220$

The general term in the expansion of $(1+a-b+c{)}^9$
$=\frac{9!}{(k_1!)(k_2!)(k_3!)(k_4!)}\left(1{)}^{k_1}\right(a{)}^{k_2}(-b{)}^{k_3}(c{)}^{k_4}$

As, $k_1+k_2+k_3+k_4=9$, so
$k_1=9-(3+4+1)=1$
So, the coefficient of $a^3{b}^{4}c$
$=\frac{9!}{(1!)(3!)(4!)(1!)}\left(a{)}^3\right(-b{)}^4(c{)}^1=\frac{9!}{(3!)(4!)}=\frac{9\cdot 8\cdot 7\cdot 6\cdot 5}{6}=2520$