Question

a swim meet has 13 contestants . the first heat has 6 simmers . how many different ways can the contestants be arranged in the first heat

Answer 1

Answer: 1, 235, 520 different arrangements
Explanation: Since we want to arrange them, we care about the order in which they come in.
Let's firstly think about it in a diagrammatic way before diving into the permutations side of things.

We have 13 swimmers, let's name them from 1 to 13. Now, we want to arrange six of them in a line (hypothetically).

Thus, we can arrange the first six people:

1 2 3 4 5 6
1 2 3 4 6 5
1 2 3 6 5 4
1 2 3 6 4 5
...

In fact, we have 6! ways in arranging six objects into six places, which is 720 different ways.

Now, let's think about it in a bigger spectrum. If we have 13 people and we want to arrange them in 13 blocks, we would have 13! ways in arranging them:

Different permutations:
Ways in arranging 13 people into 13 different lanes is given by: $13!$
Now, we want to restrict that into 6 blocks, so we can only have 6 people in it
So, we would have (13 - 6)! ways in arranging them into 6 blocks.

So, our final number of arrangements is: $\frac{13!}{(13 - 6)!} = 1 235 520$ ways.

This is also the formula for the Permutation function represented by: $^{n}P_r$, where n is the number of objects (13) and r is the number of positions (6).