Question

Number of ways in which $Rs.18$ can be distributed amongst four persons such that nobody receives less than $Rs.3$ is

• A. $4^2$
• B. $2^4$
• C. $4!$
• D. None of these

Answer 1

The correct option is D None of these

$\text{Concept: Total number of non-negative integral solution of}{x}_1+x_2+......+x_r=n\text{is}{}^{n+r-1}{C}_{r-1}$

Also, $n$ things can be distributed in $r$ groups in ${}^{n+r-1}{C}_{r-1}$ ways

Suppose $i^{th}$ person receives $Rs.x_i;i=1,2,3,4$
Then, $x_1+x_2+x_3+x_4=18$, where $x_i\ge 4$
Let $y_i=x_i-3,i=1,2,3,4$. Then,
$y_1+y_2+y_3+y_4=6$
The total number of ways is equal to number of solutions of the above equation, which is given by ${}^{6+4-1}{C}_{4-1}{=}^9{C}_3=84$.