Question

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

• A. ${}^{n+3}{C}_3$
• B. ${}^{n+2}{C}_3$
• C. ${}^{n+4}{C}_4$
• D. ${}^{n+4}{C}_3$

Answer 1

The correct option is C ${}^{n+4}{C}_4$

Let $x_5$ be such that $x_1+x_2+x_3+x_4+x_5=n$
Now we require the non - negative integral solutions of $x_1+x_2+x_3+x_4+x_5=n$
The number of required solutions ${=}^{n+5-1}{C}_{5-1}{=}^{n+4}{C}_4$