ardz24 wrote:In how many ways can you sit 8 people on a bench if 3 of them must sit together?

A. 720
B. 2,160
C. 2,400
D. 4,320
E. 40,320

Is there any strategic approach to this question?

This question could have been worded a little more clearly, but we'll take it to mean that there's enough room for all 8 people on the bench and that the 3 who wish to sit together does not change from scenario to scenario.

Call the people A, B, C, D, E, F, G, H. Say A, B, and C insist on sitting together. If we fuse those people together, we now have A-B-C, D, E, F, G, H. In other words, we'll treat A-B-C as one entity. Along with A-B-C we have 5 more people to sit, so now we want to sit a total of six entities.. There are 6! or 720 ways we can do that.

But now we have to recognize that A-B-C don't have to be seated so that A is farthest to the left andC is farthest to the right. They could sit B-A-C or C-B-A, etc. In fact, there are 3!, or 6 ways we can sit A, B, and C.

720 * 6 = 4320. The answer is D

ardz24 wrote:In how many ways can you sit 8 people on a bench if 3 of them must sit together?

A. 720
B. 2,160
C. 2,400
D. 4,320
E. 40,320

Since we are not given any names, we can denote each person with a letter:

A, B, C, D, E, F, G, H

Let's say A, B, and C must sit together; we treat [A-B-C] as a single entity, and so we have:

[A - B - C] - D - E - F - G - H

We see that we have 6 total positions, which can be arranged in 6! = 720 ways. We also can organize [A - B - C] in 3! = 6 ways.

So, the total number of ways to arrange the group is 720 x 6 = 4,320 ways.

Answer: D