Question

The greatest coefficient in the expansion of ${(a+b+c+d)}^{23}$ is

• A. $\frac{23!}{5!6!}$
• B. $\frac{23!}{5!{(6!)}^3}$
• C. $\frac{23!}{{(6!)}^3}$
• D. $\frac{23!}{{(5!)}^2{(6!)}^2}$

Answer 1

The correct option is B $\frac{23!}{5!{(6!)}^3}$

From passage, greatest coefficient in the expansion of
$(x_1+x_2+x_3......+x_k{)}^n$ is $\frac{n!}{(q!{)}^{k-r}((q+1)!{)}^r}$
Where $q$ and $r$ are the quotient and remainder when $n$ is divided by $k$
Now, here, $n=23,k=4$
$\Rightarrow n=23=5\times 4+3\Rightarrow q=5,r=3$
Hence greatest coefficient in $(a+b+c+d{)}^{23}$ is, $=\frac{23!}{(5!)(6!{)}^3}$