How many ways can n books be placed on k distinguishable she

How many ways can n books be placed on k distinguishable shelves:

a.) if the books are indistinguishable copies of the same title?

b.) if no two books are the same, and the positions of the books on the shelves matter?

Solution

Given there are \'n\' books and must be placed in \'k\' distinguishable shelves.

(a) if the books are indistinguishable copies of the same title:

There are \'n\' books which must be placed in \'k\' shelves, then the different ways of arranging the books if they are indistinguishable copies of same title are nC_k.

In nC_k ways they can be arranged. nC_k = (n!) / (n-k)! k!

Here the arrangement is based on the combinations.

(b)if no two books are the same, and the positions of the books on the shelves matter:

The number of ways n books be placed on k distinguishable shelves, if no two books are the same, and the positions of the books on the shelves matter is nP_k.

In nP_k ways they can be arranged. nP_k = n! / k!

Here it is a permutation based arrangement.