Let’s solve each problem step by step. We’ll use the formulas for permutations and combinations:
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Permutation (nPr) = n! / (n - r)!
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Combination (nCr) = n! / [r! × (n - r)!]
We’ll calculate each one carefully.
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1. For n = 3 and r = 2
Permutation:
3P2 = 3! / (3 - 2)! = 6 / 1 =
6
Combination:
3C2 = 3! / [2! × (3 - 2)!] = 6 / [2 × 1] = 6 / 2 =
3
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2. For n = 6 and r = 4
Permutation:
6P4 = 6! / (6 - 4)! = 720 / 2! = 720 / 2 =
360
Combination:
6C4 = 6! / [4! × (6 - 4)!] = 720 / [24 × 2] = 720 / 48 =
15
*(Note: You can also think of 6C4 as same as 6C2, which is easier: 6×5/2 = 15)*
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3. For n = 3 and r = 1
Permutation:
3P1 = 3! / (3 - 1)! = 6 / 2! = 6 / 2 =
3
Combination:
3C1 = 3! / [1! × (3 - 1)!] = 6 / [1 × 2] = 6 / 2 =
3
*(When r=1, permutation and combination are always equal to n)*
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4. For n = 9 and r = 2
Permutation:
9P2 = 9! / (9 - 2)! = 9 × 8 =
72
*(Because 9! / 7! = 9 × 8 × 7! / 7! = 9 × 8)*
Combination:
9C2 = 9! / [2! × (9 - 2)!] = (9 × 8) / (2 × 1) = 72 / 2 =
36
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5. For n = 5 and r = 2
Permutation:
5P2 = 5! / (5 - 2)! = 5 × 4 =
20
Combination:
5C2 = 5! / [2! × 3!] = (5 × 4) / (2 × 1) = 20 / 2 =
10
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6. For n = 7 and r = 3
Permutation:
7P3 = 7! / (7 - 3)! = 7 × 6 × 5 =
210
Combination:
7C3 = 7! / [3! × 4!] = (7 × 6 × 5) / (3 × 2 × 1) = 210 / 6 =
35
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Final Answer:
For n = 3 and r = 2: Permutation = 6, Combination = 3
For n = 6 and r = 4: Permutation = 360, Combination = 15
For n = 3 and r = 1: Permutation = 3, Combination = 3
For n = 9 and r = 2: Permutation = 72, Combination = 36
For n = 5 and r = 2: Permutation = 20, Combination = 10
For n = 7 and r = 3: Permutation = 210, Combination = 35
Parent Tip: Review the logic above to help your child master the concept of combinations and permutations worksheet with answers.
