In how many ways can 6 members of a family be assigned to 5 rooms in a hotel such that no room is empty and all 6 members are accommodated?

This problem can be solved using the concept of combinations and permutations in combinatorics. Here are the steps:

1. First, we need to ensure that no room is empty. So, we assign one family member to each room. There are 6 family members and 5 rooms, so this can be done in 6P5 ways. The permutation formula is nPr = n! / (n-r)!, where n is the total number of items, r is the number of items to choose, and "!" denotes factorial. So, 6P5 = 6! / (6-5)! = 720 ways.

2. Now, we have one family member left and 5 rooms. This family member can go into any of the 5 rooms. So, there are 5 ways to do this.

3. Therefore, the total number of ways to assign 6 family members to 5 rooms such that no room is empty is 720 * 5 = 3600 ways.