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

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

Permutations and Combinations

Solution:

Number of ways of distributing 8 identical balls in 3 distinct boxes
$=$ Number of positive integer solutions of $x_1+x_2+x_3=8$
$={ }^{8-1} C_{3-1}$
$={ }^7 C_2=\frac{7 \times 6}{2 \times 1}=21$
$\left(x+x_2+\ldots \ldots+x_r=n\right.$, number of positive integer solution equals ${ }^{n-1} C_{r-1}$ )