Question

Three travelers arrive at a town where there are four hotels; in how many ways can they take up their quarters, each at a different hotel?

Answer 1

We have to find the total number of ways in which the travelers can stay in different hotels. To do that we will find the possibilities for each traveler separately first, using the given conditions and restrictions. Then, we will multiply all the possibilities together, to obtain the total number of ways.

Complete step by step solution:
We have to find the total number of ways in which three travelers can take up quarters in four hotels without staying together.
Let us consider our three travelers as traveler 1, traveler 2, and traveler 3.
Since we have four hotels and three travelers, thus, let us first find the possible number of ways for the traveler 1.
Now, traveler 1 can take up any hotel out of the four choices, because there is no restriction. Hence, the possible number of ways for traveler 1 will be 4.
Now we will find the possible number of ways for traveler 2.
The restriction for traveler 2 is that he cannot take the hotel taken by traveler 1. Since traveler 1 has already occupied one hotel out of the four provided choices hence, only three hotels are left unoccupied now. So, traveler 2 is left with three choices now.
Finally, let us find the possible number of ways for traveler 3.
The restriction for traveler 3 is that he cannot take the hotel taken by traveler 1 or traveler 2. Since traveler 1 has already occupied one hotel out of the four provided choices, and traveler 2 has already occupied one hotel out of the three provided choices, hence, two hotels are occupied now. This leaves traveler 3 with only two unoccupied hotels. So, traveler 3 has only two choices now.
Now, we will find the total number of ways for the three travelers. For that, we will multiply the possibilities of all the travelers together.
Total number of ways $= 4 \times 3 \times 2 = 24$

Thus, the total number of ways for the three travelers will be 24 ways.

Note:
We can commit the error of confusing the logic of this question with the concept of the permutations. But this question is not of permutations because here the order of selection is not important, whereas in permutation the order matters. Here it does not matter whichever hotel any traveler takes. There is only one restriction that they should not stay together.