Problem 4 · 2015 AMC 8
Easy
Counting & Probability
careful-counting
The Centerville Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
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Answer: E — 12 arrangements.
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Hint 1 of 2
The seating breaks into two separate jobs that don't interfere: arrange the boys at the two ends, and arrange the girls in the three middle seats. Solve each job alone.
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Hint 2 of 2
When choices are independent, total = (ways for job 1) × (ways for job 2). Arranging k people in k seats has k! ways. (This is the multiplication / rule-of-product principle.)
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Approach: split into two independent jobs and multiply
- The ends are boys-only and the middle is girls-only, so the two choices never collide — handle them separately.
- Boys in the 2 end seats: 2! = 2 ways. Girls in the 3 middle seats: 3! = 6 ways.
- By the rule of product, total = 2 × 6 = 12.
- Why this transfers: any time a setup divides into independent stages, multiply the counts. The only trap is making sure the stages really don't affect each other — here they can't, since boys and girls occupy disjoint seats.
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