Permutations and Combinations Notes & Worksheets - Free Printable

Let’s go through each problem one by one. We’ll figure out whether it’s a permutation or combination, and then calculate the answer.

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1. 3 applicants picked from a job set of 20 total applicants.

This is about choosing people — order doesn’t matter (you’re just picking who gets hired, not ranking them). So this is a combination.

Formula: C(n, r) = n! / [r!(n - r)!]

C(20, 3) = 20! / (3! × 17!)
= (20 × 19 × 18) / (3 × 2 × 1)
= 6840 / 6
= 1140

✔ Answer: Combination → 1140 ways

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2. Permutation or Combination? 5 candidates for a tutor.

Wait — the question says “5 candidates for a tutor” but doesn’t say what we’re doing with them. Are we picking one? All five? Ordering them?

Looking at context — probably means selecting 1 tutor from 5 candidates. If you’re just choosing 1 person, order doesn’t matter → Combination.

But if you were assigning roles like “first choice”, “second choice”, etc., that would be permutation. Since it just says “for a tutor”, likely choosing 1 → Combination.

C(5,1) = 5

✔ Answer: Combination → 5 ways

*(Note: If the problem meant something else, like arranging all 5 in order, it would be permutation. But based on wording, we assume selecting 1.)*

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3. How many different ways to arrange the letters in HONOR?

“Arrange” means order matters → Permutation.

But note: HONOR has repeated letters — two O’s.

Total letters: 5
Repeated: O appears twice.

So permutations = 5! / 2! = 120 / 2 = 60

✔ Answer: Permutation → 60 ways

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4. 5 cards drawn from a deck of 52.

Drawing cards — usually, unless specified otherwise, order doesn’t matter (like in poker hands). So this is a combination.

C(52, 5) = 52! / (5! × 47!)
= (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1)
Let’s compute step by step:

Numerator: 52 × 51 = 2652
2652 × 50 = 132600
132600 × 49 = 6,497,400
6,497,400 × 48 = 311,875,200

Denominator: 5! = 120

Now divide: 311,875,200 ÷ 120 = ?

First, 311,875,200 ÷ 10 = 31,187,520
Then ÷ 12 = 31,187,520 ÷ 12

Better way: Divide numerator and denominator step by step.

We can simplify before multiplying:

(52/4) × (51/3) × (50/2) × 49 × 48 / (5×1) → wait, better to do:

Standard value: C(52,5) = 2,598,960

✔ Answer: Combination → 2,598,960 ways

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5. Pick 2 dogs from a litter of 8. How many combinations?

Order doesn’t matter → Combination

C(8,2) = (8 × 7) / (2 × 1) = 56 / 2 = 28

✔ Answer: Combination → 28 ways

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6. Arrange 10 students on stage for a choral concert.

“Arrange” → order matters → Permutation

P(10,10) = 10! = 3,628,800

✔ Answer: Permutation → 3,628,800 ways

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7. Want 4 scoops of ice cream. There are 15 flavors. You cannot repeat flavor. How many combinations?

Choosing 4 different flavors, no repeats, order doesn’t matter (unless specified — but it says “combinations”) → Combination

C(15,4) = 15! / (4! × 11!)
= (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1)
= (32760) / 24
= 1365

✔ Answer: Combination → 1,365 ways

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8. How many different locker combinations are there for a 3-digit code? Numbers can be used more than once.

Locker “combinations” here actually mean sequences where order matters AND repetition allowed → This is permutation with repetition, not mathematical combination.

Each digit: 0-9 → 10 choices per digit.

3 digits → 10 × 10 × 10 = 1000

Even though it says “combinations”, in real life lockers care about order → so it’s permutation-style counting.

✔ Answer: 1000 possible codes

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9. Family dinner: chicken, pasta, rolls, corn, pie. You want 3 things. How many combos if you can get same thing more than once?

You can repeat items → this is combinations with repetition.

Formula: C(n + r - 1, r) = C(5 + 3 - 1, 3) = C(7,3)

C(7,3) = (7 × 6 × 5) / (3 × 2 × 1) = 210 / 6 = 35

✔ Answer: 35 combinations

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10. Number of possible outcomes from choosing 5 applicants out of 30 applicants total.

Again, choosing people — order doesn’t matter → Combination

C(30,5) = 30! / (5! × 25!)
= (30 × 29 × 28 × 27 × 26) / (5 × 4 × 3 × 2 × 1)

Calculate numerator:
30 × 29 = 870
870 × 28 = 24,360
24,360 × 27 = 657,720
657,720 × 26 = 17,100,720

Denominator: 120

Divide: 17,100,720 ÷ 120 = ?

17,100,720 ÷ 10 = 1,710,072
÷ 12 = 142,506

Wait — let me check standard value: C(30,5) = 142,506

✔ Answer: Combination → 142,506 ways

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Final Answers Summary:

1. Combination → 1140
2. Combination → 5
3. Permutation → 60
4. Combination → 2,598,960
5. Combination → 28
6. Permutation → 3,628,800
7. Combination → 1,365
8. 1000 (permutation with repetition)
9. 35 (combination with repetition)
10. Combination → 142,506

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Final Answer:
1. 1140
2. 5
3. 60
4. 2,598,960
5. 28
6. 3,628,800
7. 1,365
8. 1000
9. 35
10. 142,506