Permutations worksheets

Seven runners and three medals make 210 possible results. These free printable worksheets count the ways to fill some places from a larger group when each place is different, with an answer sheet.

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Permutations

The answer key prints on a separate page.

What's on this sheet

Every story here has places that differ: gold or silver, president or treasurer, front seat or back. So there are 7 choices for the first medal, then 6 for the next, then 5, and the count stops after three places because only three medals are given. That stopping point is the new idea after arranging a whole group.

Why this sheet works

A common slip is to multiply all the way down to 1, as if everyone got a place. Writing one blank for each place before filling in the numbers stops that, because the blanks show how many factors there should be. Three medals means three numbers.

How this one works

8 runners race for 3 different medals. In how many ways can the medals be won?

876

8 × 7 × 6=336

There are 8 choices for the first place. Whoever took it cannot take the next, so that place has 7, and so on for 3 places.
8 × 7 × 6 = 336.

Permutations and combinationsPermutations

Name: Date:

Order matters here. Each place has one fewer choice than the last.

8 runners race for 3 different medals. In how many ways can the medals be won?

9 runners race for 4 different medals. In how many ways can the medals be won?

4 dogs compete for the top 2 places. In how many ways can those places be filled?

9 students take the 4 seats in the front row, one to a seat. In how many ways can they sit?

A club of 5 students picks 2 officers, each with a different job. In how many ways can it do that?

4 runners race for 3 different medals. In how many ways can the medals be won?

10 students take the 4 seats in the front row, one to a seat. In how many ways can they sit?

5 dogs compete for the top 2 places. In how many ways can those places be filled?

Permutations and combinations · PermutationsAnswer key

1. 8 runners race for 3 different medals. In how many ways can the medals be won? → 8 × 7 × 6 = 3362. 9 runners race for 4 different medals. In how many ways can the medals be won? → 9 × 8 × 7 × 6 = 30243. 4 dogs compete for the top 2 places. In how many ways can those places be filled? → 4 × 3 = 124. 9 students take the 4 seats in the front row, one to a seat. In how many ways can they sit? → 9 × 8 × 7 × 6 = 30245. A club of 5 students picks 2 officers, each with a different job. In how many ways can it do that? → 5 × 4 = 206. 4 runners race for 3 different medals. In how many ways can the medals be won? → 4 × 3 × 2 = 247. 10 students take the 4 seats in the front row, one to a seat. In how many ways can they sit? → 10 × 9 × 8 × 7 = 50408. 5 dogs compete for the top 2 places. In how many ways can those places be filled? → 5 × 4 = 20

Permutations worksheets

What's on this sheet

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