Three men and two women are seated in a row of 6 chairs in an auditorium. If the seats at either end of the row are occupied by tom and jerry, how manu different ways can the five people be seated?

6C3+6C2

= (6*5*4)/(2*3) + (6*5)/(2)
= 20+15
=35

Oh, I see we have Tom and Jerry securing the best seats in the house! Let's calculate the number of ways the remaining five people can be seated.

Since Tom and Jerry are already seated at the ends, we have 4 chairs left for the other five people. The order of seating matters, so we can use permutations to calculate the number of arrangements.

The number of ways to arrange 5 people in 4 chairs is 4P5, which can be calculated as:

4P5 = 4! / (4-5)! = 4 x 3 x 2 x 1 / (-1)! = 24

So, there are 24 different ways to seat the remaining five people. I hope they bring popcorn and enjoy the show!

To solve this problem, we can consider the cases when Tom and Jerry sit on the left end, and when they sit on the right end.

Case 1: Tom and Jerry sit on the left end.
In this case, there are 4 remaining people (2 men and 2 women) to be seated in the remaining 4 chairs. We can calculate the number of ways to arrange the remaining people using the permutation formula "nPr", where n is the total number of items and r is the number of items to be selected. Therefore, the number of ways to arrange the remaining people is 4P4 = 4! = 24.

Case 2: Tom and Jerry sit on the right end.
Similarly, in this case, there are 4 remaining people (2 men and 2 women) to be seated in the remaining 4 chairs. Using the permutation formula, the number of ways to arrange the remaining people is 4P4 = 4! = 24.

Since these cases are mutually exclusive (both cannot occur simultaneously), the total number of ways to arrange the five people is obtained by adding the results from the two cases:
Total number of ways = 24 + 24 = 48.

Therefore, there are 48 different ways the five people can be seated.

To find the number of different ways the five people can be seated, we need to consider the following points:

1. Since Tom and Jerry have fixed seats at either end of the row, we can consider them as one block. So, we now have four seats remaining for the other three men and two women.

2. The number of ways these remaining four seats can be filled by the three men is given by the permutation formula: P(4,3) = 4! / (4-3)! = 4! / 1! = 4 x 3 x 2 = 24.

3. Similarly, the number of ways the two women can be seated in the remaining two seats is given by the permutation formula: P(2,2) = 2! / (2-2)! = 2! / 0! = 2 x 1 = 2.

4. To find the total number of different ways the five people can be seated, we multiply the number of ways the three men can be seated (24) by the number of ways the two women can be seated (2). So, the total number of different ways is 24 x 2 = 48.

Therefore, there are 48 different ways the five people can be seated in the row of 6 chairs, with Tom and Jerry already seated at the ends.