In how many ways can 8 identical balls be dis | vedclass

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(B) We need to distribute $8$ identical balls into $3$ distinct boxes such that each box contains at least $1$ ball. This is equivalent to finding the

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$21$

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(B) We need to distribute $8$ identical balls into $3$ distinct boxes such that each box contains at least $1$ ball. This is equivalent to finding the number of positive integer solutions to the equation $x_1 + x_2 + x_3 = 8$,where $x_i \ge 1$.Using the stars and bars formula,the number of ways is given by $\binom{n-1}{r-1}$,where $n=8$ and $r=3$.Number of ways = $\binom{8-1}{3-1} = \binom{7}{2}$.Calculating the value: $\binom{7}{2} = \frac{7 	imes 6}{2 	imes 1} = 21$.

In how many ways can $8$ identical balls be distributed into $3$ distinct boxes such that no box remains empty?

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