Question

The total number of non-negative integral solutions of $x_1+x_2+x_3+x_4=100$ is

• A. ${{}^{100}C}_3$
• B. ${{}^{99}C}_3$
• C. ${{}^{100}C}_4$
• D. ${{}^{99}C}_4$

Answer 1

The correct option is A ${{}^{100}C}_3$

$x_1+x_2+x_3+x_4=100$

Total number of non-negative integral solutions of the given equations is same as the number of ways of distributing $100$ items among $4$ persons such that each person can receive any number of items.

Hence, total number of solutions ${=}^{100+4-1}{C}_{4-1}={}^{103}{C}_3$