Question

The number of distinct terms in the expansion of $(x+2y-3z+5w-7u{)}^n$ is :

• A. $n+1$
• B. ${}^{n+4}{C}_4$
• C. ${}^{n+4}{C}_n$
• D. $\frac{(n+1)(n+2)(n+3)(n+4)}{24}$

Answer 1

Answer: (B), (C), (D)

${}^{n+4}{C}_4$
${}^{n+4}{C}_n$
$\frac{(n+1)(n+2)(n+3)(n+4)}{24}$

Number of terms in the given expansion is nothing but the non-negative integral solutions of the equation:

$a+b+c+d+e=20$,

where $a,b,c,d,e$ are powers of $x,y,z,w,u$ respectively.

Total number of non-negative integral solutions will be

${{}^{n+5-1}C}_{5-1}={{}^{n+4}C}_4$

$\Rightarrow {{}^{n+4}C}_4=\frac{(n+4)(n+3)(n+2)(n+1)}{4\times 3\times 2\times 1}$

Number of distinct terms in the given expansion will be

$\frac{(n+4)(n+3)(n+2)(n+1)}{24}$