Permutations and combinations practice worksheet with 12 problems.

Let’s solve each problem step by step. We’ll use the formulas for permutations and combinations:

- Permutation: P(n, r) = n! / (n - r)!
→ Order matters. Example: arranging people in a line.

- Combination: C(n, r) = n! / [r! × (n - r)!]
→ Order does NOT matter. Example: picking a team from a group.

We’ll compute each one carefully.

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1) –C(20, 18)

C(20, 18) = 20! / [18! × (20 - 18)!] = 20! / (18! × 2!)
But note: C(n, r) = C(n, n-r), so C(20, 18) = C(20, 2)

C(20, 2) = (20 × 19) / (2 × 1) = 380 / 2 = 190

So –C(20, 18) = –190

✔ Answer: –190

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2) C(6, 5) × P(9, 1) / 3

First, C(6, 5) = C(6, 1) = 6
P(9, 1) = 9! / (9 - 1)! = 9! / 8! = 9

So numerator: 6 × 9 = 54
Divide by 3: 54 / 3 = 18

✔ Answer: 18

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3) 8 – P(3, 3) – P(2, 2)

P(3, 3) = 3! / 0! = 6 / 1 = 6
P(2, 2) = 2! / 0! = 2 / 1 = 2

So: 8 – 6 – 2 = 0

✔ Answer: 0

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4) C(3, 1) / P(4, 2)

C(3, 1) = 3
P(4, 2) = 4 × 3 = 12

So: 3 / 12 = 1/4

✔ Answer: 1/4

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5) P(8, 2) / [C(5, 3) + 6]

P(8, 2) = 8 × 7 = 56
C(5, 3) = C(5, 2) = (5 × 4)/2 = 10
Denominator: 10 + 6 = 16

So: 56 / 16 = 3.5 or 7/2

✔ Answer: 7/2 (or 3.5 — but fraction is better unless specified)

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6) 4! + C(61, 60)

4! = 24
C(61, 60) = C(61, 1) = 61

So: 24 + 61 = 85

✔ Answer: 85

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7) 3P(6, 4)

P(6, 4) = 6 × 5 × 4 × 3 = 360
Multiply by 3: 3 × 360 = 1080

✔ Answer: 1080

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8) –P(9, 3) + C(17, 14)

P(9, 3) = 9 × 8 × 7 = 504 → so –504
C(17, 14) = C(17, 3) = (17 × 16 × 15) / (3 × 2 × 1) = (4080) / 6 = 680

So: –504 + 680 = 176

✔ Answer: 176

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9) C(8, 7) – P(5, 1)

C(8, 7) = C(8, 1) = 8
P(5, 1) = 5

So: 8 – 5 = 3

✔ Answer: 3

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10) –C(4, 3) + P(7, 4)

C(4, 3) = C(4, 1) = 4 → so –4
P(7, 4) = 7 × 6 × 5 × 4 = 840

So: –4 + 840 = 836

✔ Answer: 836

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11) 2! – 11 / C(5, 5)

2! = 2
C(5, 5) = 1
So: 11 / 1 = 11

Then: 2 – 11 = –9

⚠️ Wait — order of operations! Is it 2! – (11 / C(5,5)) or (2! – 11) / C(5,5)?

Looking at how it's written: “2! – 11 / C(5, 5)” — division comes before subtraction.

So yes: 2! – (11 / C(5,5)) = 2 – (11 / 1) = 2 – 11 = –9

✔ Answer: –9

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12) 6C(7, 5) + 5P(9, 2)

C(7, 5) = C(7, 2) = (7 × 6)/2 = 21 → 6 × 21 = 126
P(9, 2) = 9 × 8 = 72 → 5 × 72 = 360

Total: 126 + 360 = 486

✔ Answer: 486

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Final Answer:
1) –190
2) 18
3) 0
4) 1/4
5) 7/2
6) 85
7) 1080
8) 176
9) 3
10) 836
11) –9
12) 486