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Combinations worksheets

A team of 3 from 7 students can be picked in 35 ways, not 210, because a team has no first or second. These free printable worksheets count groups where order does not matter, with an answer sheet.

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Combinations
Combinations worksheet — Order does not matter, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The method has two steps and the second one is the point. First count as if order did matter: 7 × 6 × 5 = 210. That number has counted every team several times, once for each way its three people could be lined up. Three people can be lined up in 3 × 2 × 1 = 6 ways, so the real count is 210 ÷ 6 = 35.

Why this sheet works

Why divide? Because Ann, Ben and Cal is the same team as Cal, Ann and Ben, and the first count treated them as different. Listing the six orders of one small team, once, usually makes this clear in a way the formula does not.

How this one works

A coach picks 3 of 5 players to start the game. How many different groups are possible?

543
5 × 4 × 3=60
60÷6=10
  1. Start as if order mattered. There are 5 choices for the first place, then 4 for the next, and so on for 3 places.
  2. 5 × 4 × 3 = 60.
  3. But a group has no first or second. The same 3 can be ordered in 3 × 2 × 1 = 6 ways, and all of those are one group. So divide: 60 ÷ 6 = 10.

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