Question

The total number of ways in which $15$ identical blankets can be distributed among four persons so that each of them gets at least two blankets is equal to

• A. ${}^{10}{C}_3$
• B. ${}^9{C}_3$
• C. ${}^{11}{C}_3$
• D. None of these

Answer 1

The correct option is A ${}^{10}{C}_3$

$\text{Concept: Total number of non-negative integral solution of}{x}_1+x_2+......+x_r=n\text{is}{}^{n+r-1}{C}_{r-1}$
Also, $n$ identical things can be distributed in $r$ groups in ${}^{n+r-1}{C}_{r-1}$ ways

Let the blankets received by the persons are $x_1,x_2,x_3$ and $x_4$. We have,
$x_1+x_2+x_3+x_4=15$ and $x_i\ge 2$
$\Rightarrow (x_1-2)+(x_2-2)+(x_3-2)+(x_4-2)=7$
$\Rightarrow y_1+y_2+y_3+y_4=7$ (where $y_i=x_i-2\ge 0)$
The required number is equal to the number of non-negative integral solutions of this equation which is equal to ${}^{4+7-1}{C}_7$, i.e., ${}^{10}{C}_7{=}^{10}{C}_3$.