ACT Preparation / Unit 15 ACT 1431 worked example, 60 to 90 seconds
Permutation or Combination
The idea
A counting problem is solved in three steps. First, decide whether order matters: titles, positions, rankings and sequences mean it does; groups, teams and committees mean it does not. Second, identify n, the pool, and r, the number chosen. Third, compute nPr if order matters and nCr if not, or use the multiplication principle when the choices come from different pools.
In practice
Worked example: in a race of 8 runners, gold, silver and bronze can be awarded in 8P3 = 8 × 7 × 6 = 336 ways. A coach choosing 3 of the 8 for a relay pool, with roles not yet set, has 8C3 = 336/6 = 56 ways. A lunch of one of 4 sandwiches and one of 3 drinks has 4 × 3 = 12 ways.
Where people slip
The combination count is the permutation count divided by r!, the ways to order the chosen items. When two answer choices differ by exactly that factor, the question turns on whether order matters.
Your place on the trail
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