Let’s solve each problem step by step. We’re dealing with
combinations — that means the order doesn’t matter. For example, picking Alice then Bob is the same as picking Bob then Alice when forming a committee.
The formula for combinations is:
> C(n, r) = n! / [r! × (n - r)!]
Where:
- n = total number of items
- r = number we are choosing
- “!” means factorial (e.g., 5! = 5×4×3×2×1)
But you don’t need to calculate factorials every time — your calculator or textbook probably has a “nCr” button. We’ll use that idea here.
---
Problem 1:
A company needs to select 3 people from 7 applicants. How many different ways?
This is C(7, 3).
C(7, 3) = 7! / (3! × 4!)
= (7×6×5) / (3×2×1)
= 210 / 6
=
35
✔ Answer:
35
---
Problem 2:
Select 4 people from 9 applicants.
C(9, 4) = 9! / (4! × 5!)
= (9×8×7×6) / (4×3×2×1)
= 3024 / 24
=
126
✔ Answer:
126
---
Problem 3:
Committee of 5 members from 12 employees.
C(12, 5) = 12! / (5! × 7!)
= (12×11×10×9×8) / (5×4×3×2×1)
= 95040 / 120
=
792
✔ Answer:
792
---
Problem 4:
Choose 3 students from 10 for a project.
C(10, 3) = 10! / (3! × 7!)
= (10×9×8) / (3×2×1)
= 720 / 6
=
120
✔ Answer:
120
---
Problem 5:
Form a team of 4 from 8 players.
C(8, 4) = 8! / (4! × 4!)
= (8×7×6×5) / (4×3×2×1)
= 1680 / 24
=
70
✔ Answer:
70
---
Problem 6:
Pick 2 books from 6 on a shelf.
C(6, 2) = 6! / (2! × 4!)
= (6×5) / (2×1)
= 30 / 2
=
15
✔ Answer:
15
---
Problem 7:
Select 3 flavors from 8 ice cream flavors.
C(8, 3) = 8! / (3! × 5!)
= (8×7×6) / (3×2×1)
= 336 / 6
=
56
✔ Answer:
56
---
Problem 8:
Choose 4 songs from 10 for a playlist.
C(10, 4) = 10! / (4! × 6!)
= (10×9×8×7) / (4×3×2×1)
= 5040 / 24
=
210
✔ Answer:
210
---
Problem 9:
Pick 5 questions from 12 on an exam.
C(12, 5) = same as Problem 3 →
792
✔ Answer:
792
---
Problem 10:
Select 3 toppings from 7 available.
C(7, 3) = same as Problem 1 →
35
✔ Answer:
35
---
Final Answer:
1. 35
2. 126
3. 792
4. 120
5. 70
6. 15
7. 56
8. 210
9. 792
10. 35
Parent Tip: Review the logic above to help your child master the concept of permutations and combinations worksheet with answers.
