Question

There are 12 blackboards that are to be distributed to 8 different schools. Assume that all 12 blackboards are indistinguishable. In how many different ways can these blackboards be distributed so that there must be at least one board for each school?

Answer 1

330

Answer 2

This problem requires distributing $n=12$ indistinguishable blackboards to $k=8$ distinguishable schools such that each school receives at least one blackboard. This is equivalent to finding the number of positive integer solutions to the equation $x_1 + x_2 + \dots + x_8 = 12$, where $x_i$ is the number of blackboards given to school $i$.

Using the stars and bars technique for positive integer solutions, the number of ways is given by the formula $\binom{n-1}{k-1}$.

Applying this formula: Number of ways = $\binom{12-1}{8-1} = \binom{11}{7}$

Calculating the binomial coefficient: $\binom{11}{7} = \binom{11}{11-7} = \binom{11}{4}$ $\binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}$ $\binom{11}{4} = 11 \times \frac{10}{2} \times \frac{9}{3} \times \frac{8}{4} = 11 \times 5 \times 3 \times 2$ $\binom{11}{4} = 330$