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Arranging things in order worksheets

Five books can stand on a shelf in 120 different orders, which surprises most people the first time. These free printable worksheets count the ways to arrange a whole group in a line, with an answer sheet.

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Arranging things in order
Arranging things in order worksheet — Groups of 3 to 7, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The reasoning goes one place at a time. The first spot on the shelf could hold any of the 5 books. Whichever went there cannot go in the next spot, so that one has 4 choices, then 3, then 2, then 1. Multiplying them gives 5 × 4 × 3 × 2 × 1 = 120.

Why this sheet works

This is the counting principle with one change: the list shrinks as it is used. On a shirts-and-shorts question the choices are independent, and picking a shirt leaves every pair of shorts available. Here each choice uses something up. Saying that difference out loud makes the next sheets, permutations and combinations, much easier to follow.

How this one works

7 runners finish a race with no ties. In how many different orders can they finish?

7654321
7 × 6 × 5 × 4 × 3 × 2 × 1=5040
  1. There are 7 choices for the first place. Whoever took it cannot take the next, so that place has 6, and so on for 7 places.
  2. 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.

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