The number of ways of distributing 8 identical balls in 3 distinct boxes, so that none of the boxes is empty, is

Question

The number of ways of distributing 8 identical balls in 3 distinct boxes, so that none of the boxes is empty, is (A) 5 (B) 21 (C) 38 (D)  8 C3

Tardigrade Admin · Accepted Answer

Number of ways of distributing 8 identical balls in 3 distinct boxes

=

Number of positive integer solutions of

x_{1} + x_{2} + x_{3} = 8

=^{8 - 1} C_{3 - 1}

=^{7} C_{2} = \frac{7 \times 6}{2 \times 1} = 21

(x + x_{2} + \dots\dots + x_{r} = n

, number of positive integer solution equals

^{n - 1} C_{r - 1}

)

Q. The number of ways of distributing 8 identical balls in 3 distinct boxes, so that none of the boxes is empty, is