Question

Find the greatest coefficient in the expansion of ${(a+b+c+d)}^{15}$

• A. $\frac{5!14!}{{(4!)}^3}$
• B. $\frac{15!14!}{(3!){(4!)}^3}$
• C. $\frac{15!}{(3!2!){(4!)}^3}$
• D. $\frac{15!}{(3!){(4!)}^3}$

Answer 1

The correct option is D $\frac{15!}{(3!){(4!)}^3}$

We know that, 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, $n=15,k=4$
$\Rightarrow n=15=3\times 4+3\Rightarrow q=3,r=3$
Hence greatest coefficient in $(a+b+c+d{)}^{15}$ is, $=\frac{15!}{(3!)(4!{)}^3}$