Question

$10$ apples are distributed at random among $6$ persons. The probability that at least one of them will receive none is

• A. $\frac{6}{143}$
• B. $\frac{{}^{14}{C}_4}{{}^{15}{C}_5}$
• C. $\frac{137}{143}$
• D. none of these

Answer 1

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

10 apples can be distributed among 6 people (without any constraint) in ${}_{(6-1)}^{(10+6-1)}C{=}_5^{15}C=3003$ ways
10 apples can be distributed among 6 people so that each gets at least one apple in ${}_{(6-1)}^{(10-1)}C{=}_5^{9}C=126$ ways
Therefore, probability that at least one of the people gets none

= $\frac{3003-126}{3003}=\frac{2877}{3003}=\frac{137}{143}$