The number of wasts of choosing 4 people from 12 is the same as choosing 8 people from 12. How come?

Question

The number of ways of choosing 4 people from 12 is the same as the number of ways as choosing 8 people from 12 people. I don't understand this. Can someone explain why these tasks can be done in the same number of ways?

My logic just don't understand this, but I do know it is true.

Answer 1

Just think of breaking them into two groups. In one group you select 4, the other group is the remaining 8. Notice how the way you choose 4 for the first group, is the same as the ways to pick 8 to have in the other group.

Let's take a simpler case of 3 people. The ways to pick one person is 3 (A, B or C), but if you think of the people left (AC, AC, AB) there are also 3 ways to pick 2 people to leave behind.

There is a one-to-one correspondence between the groups.

Pick A, Leave BC <--> Pick BC, Leave A
Pick B, Leave AC <--> Pick AC, Leave B
Pick C, Leave AB <--> Pick AB, Leave C

Answer 2

Either way, you're splitting a group of 12 into a group of 4 on one side of the room and a group of 8 on the other side of the room.

Picking a group of 4 and picking a group of 8 is doing EXACTLY THE SAME THING.

That's as long as each group counts as one, period. If you're doing permutations, and counting each group multiple times as you rearrange its members, that's a different matter.