Question

The total number of non negative integral solution of the inequality ${\sum }_{i=1}^4{x}_i\le 100$

• A. ${}^{103}{\text{C}}_3$
• B. ${}^{104}{\text{C}}_4$
• C. ${}^{103}{\text{C}}_4$
• D. ${}^{104}{\text{C}}_3$

Answer 1

Answer: (B)

${}^{104}{\text{C}}_4$

Number of non negative solution of $x_1+x_2+x_3+x_4\le 100$ will be same as the number of non negative solution of the equation
$x_1+x_2+x_3+x_4+x_5=100(\ast )$
and the number of solutions of the equation is equal to the number of term in the expansion of
${(x_1+x_2+...+x_5)}^{100}$ and the number of terms are ${}^{100+5-1}{C}_{5-1}{=}^{104}{C}_4$
By using Number of terms in ${(x_1+x_2+...+x_r)}^n{=}^{n+r-1}{C}_{r-1}$