Discussion
The no. of ways to distribute n distinguishable objects into k distinguishable boxes, so that \(n_i\) objects are placed into box i, i = 1, 2, …. k equals which of the following ?
Answer: C
The number of ways to distribute n distinguishable objects into k distinguishable boxes so that box i gets \(n_i\) objects is the multinomial coefficient
\[ \frac{n!}{n_1!\, n_2! \cdots n_k!} \]The reasoning is that the n objects can be arranged in n! orders, and the order inside each box does not matter, so the count is divided by \(n_i!\) for each box. The options with sums of factorials in the numerator or denominator have no such combinatorial meaning. The keyed answer is the third option.
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