The host of a dinner party must determine how to seat himself and 5 gu

Since he always chooses the same seat, the remaining 5 guests can be seated in 5! = 120 ways.

Note: The wording of the question is confusing, and you might think that the answer should be 6! = 720.

The fine distinction between the incorrect solution and the correct one is that, while there are 6 individuals to be seated, we don’t consider the host as one of them. Here are some examples that illustrate the phrase “the host always chooses the same seat for himself.”

Example 1: The host chooses the first seat for himself. Thus, there are 5! = 120 ways to seat his guests.

Example 2: The host chooses the second seat for himself. Thus, there are 5! = 120 ways to seat his guests.

We could illustrate this with 4 more examples. In them, the host chooses the 3rd, 4th, 5th, or 6th seat for himself. And in each case, his guests can be seated in 5! = 120 different ways.

We see that, because of the wording of the question, we don’t consider the host’s choice of a seat. Rather, we concentrate on the number of ways he can seat his guests, once he has chosen his seat.

Answer: D