How many ways can 4 finalists be selected from 50 people?

Answer 1

You have 50 ways to select the first.

For each of those 50, you have 49 ways to select the second.

Only 49 ways, because for each of the 50, there are only 49 left to select from.

That's 50 x 49 ways to select the first and the second.

For each of those 50 x 49 ways, you have 48 ways to select the third.

That's 50 x 49 x 48 ways to select the first, second and third.

For each of those 50 x 49 x 48 ways, you have 47 ways to select the fourth.

That's 50 x 49 x 48 x 47 ways to select the first, second and third and fourth.

But if it doesn't matter what order they are selected in, the same four people have been counted multiple times. For example, if people A, B, C and D have been selected, we will have counted A, B, C and D as one way, and D, C, B and A as another, B, D, A and D as another. In fact, there are 4 x 3 x 2 x 1 ways of counting any four people.

So if order does not matter, there are (50 x 49 x 48 x 47) / (4 x 3 x 2 x 1) ways.

How many ways can 5 finalists be selected from 100 people?