Question

If $10$ identical apples are distributed among $6$ persons at random then the probability that at least one of them will receive none is

• A. $\frac{6}{143}$
• B. $\frac{{}^{14}{C}_4}{{}^3{C}_3}$
• C. $\frac{137}{143}$
• D. $\frac{1}{143}$

Answer 1

The correct option is C $\frac{137}{143}$

$10$ apples can be distributed among $6$ people in ${}^{(10+6-1)}{C}_{(6-1)}{=}^{15}{C}_5$
$10$ apples can be distributed among $6$ people so that each gets at least one apple in ${}^{(10-1)}{C}_{(6-1)}{=}^9{C}_5$
Hence, required probability $=\frac{{}^{15}{C_5}^9{C}_5}{{}^{15}{C}_5}=\frac{137}{143}$