Question

How many ways can 6 people sit on a bench if 3 of them have to sit side by side?

Answer 1

How many different ways can you arrange 4 objects? One of those objects is special, and contains 3 sub-objects. How many ways can it be arranged? How do these answers combine to get the final answer?

Answer 2

There are six ways to arrange a group of three people such that they are still sitting together:

abc, bac, cab, acb, bca, cba

Then you need to multiply that 6 by the ways you can seat the other three people around that group. You can have them all on the left of the first group of three (6 different permutations of people to achieve that), all on the right side (another six ways) two on the left and one on the right (six more ways) and one on the left and two on the right (a final six ways). So 6 x (6+6+6+6) is 6 x 24 = 144

Answer 3

You got the first part right, you can treat the 3 as one. But how many ways are there for the 3 to sit side by side? 3! as well. So you have 4! for the 1st part and another 3! for considering the three sitting next to each other.
That gives 4!×3! =144.

Answer 4

That the three can be rearranged within their group, in 3! different ways.

So multiply the number of arrangements of 4 objects by 3!.

Answer 5

It's not really just one generic unit though, it's a special unit that occurs in multiple configurations within an arrangement with the other, more typical units.

Answer 6

Because the 3 people need to sit side by side consider them as one unit therefore there would be 4! ways of arranging all the people and within the 3 people unit there would be 3! Ways of arranging them so together the total number of ways of arranging them is 4!x3! Ways
I hope this is a more intuitive understanding for you

Answer 7

4×3!×3!=144