Question

The number of non-negative integral solution of $x_1+x_2+x_3+x_4\le n$ ( where $n$ is a positive integer ) is

• A. ${}^{n+4}{C}_n$
• B. ${}^{n+4}{C}_4$
• C. ${}^{n+3}{C}_3$
• D. ${}^{n+3}{C}_n$

Answer 1

Answer: (A), (B)

The correct options are
A ${}^{n+4}{C}_n$
B ${}^{n+4}{C}_4$

Given: $x_1+x_2+x_3+x_4\le n$

Let $x_5$ be such that $x_1+x_2+x_3+x_4+x_5=n$

We now seek the non-negative integral solutions of $x_1+x_2+x_3+x_4+x_5=n$.

The required number of ways is ${{}^{n+4}C}_n={{}^{n+4}C}_4$