Here are the step-by-step solutions for each problem on the worksheet. Since order matters in all these questions (being Captain is different from being Secretary, 1st place is different from 2nd place, etc.), we use
permutations.
1. Ski Club Officers
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The Setup: There are 10 members and 3 specific positions (Captain, Co-captain, Secretary).
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Step 1: Choose the Captain. There are
10 choices.
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Step 2: Choose the Co-captain. One person is already Captain, so there are
9 choices left.
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Step 3: Choose the Secretary. Two people are taken, so there are
8 choices left.
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Calculation: $10 \times 9 \times 8 = 720$
2. Board of Directors
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The Setup: There are 10 members and 4 specific positions (President, VP, Secretary, Treasurer).
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Step 1: President:
10 choices.
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Step 2: VP:
9 choices.
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Step 3: Secretary:
8 choices.
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Step 4: Treasurer:
7 choices.
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Calculation: $10 \times 9 \times 8 \times 7 = 5,040$
3. Radio Show Songs
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The Setup: There are 8 songs available, and the DJ needs to play 4 in a specific order.
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Step 1: First song:
8 choices.
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Step 2: Second song:
7 choices.
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Step 3: Third song:
6 choices.
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Step 4: Fourth song:
5 choices.
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Calculation: $8 \times 7 \times 6 \times 5 = 1,680$
4. Ranking Movies
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The Setup: You saw 10 movies and need to rank the top 3 (1st, 2nd, and 3rd best).
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Step 1: Best movie:
10 choices.
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Step 2: Second best:
9 choices.
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Step 3: Third best:
8 choices.
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Calculation: $10 \times 9 \times 8 = 720$
5. Air Race Finishers
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The Setup: There are 6 planes, and we need to find the number of ways the top 3 can finish (1st, 2nd, 3rd).
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Step 1: 1st place:
6 choices.
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Step 2: 2nd place:
5 choices.
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Step 3: 3rd place:
4 choices.
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Calculation: $6 \times 5 \times 4 = 120$
6. Casting Roles
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The Setup: There are 8 actors and 3 specific roles (Danny, Kenickie, Marty).
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Step 1: Cast Danny:
8 choices.
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Step 2: Cast Kenickie:
7 choices.
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Step 3: Cast Marty:
6 choices.
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Calculation: $8 \times 7 \times 6 = 336$
7. Concert Lineups
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The Setup: There are 7 bands, but only 4 can play. The order they play in (the lineup) matters.
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Step 1: First band:
7 choices.
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Step 2: Second band:
6 choices.
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Step 3: Third band:
5 choices.
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Step 4: Fourth band:
4 choices.
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Calculation: $7 \times 6 \times 5 \times 4 = 840$
──────────────────────────────────────
Final Answer:
1. 720
2. 5,040
3. 1,680
4. 720
5. 120
6. 336
7. 840
Parent Tip: Review the logic above to help your child master the concept of permutation and combination worksheet.
