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Math 30-1 · Worksheets

Counting · Set B

Ten questions of mixed difficulty, covering Counting. Print it, or work through it on screen — the answer key starts on its own page.

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Counting · Set B

Math 30-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Using the letters A, B, C, D, E exactly once each, how many arrangements have no two vowels next to each other?

    1. a) 120
    2. b) 36
    3. c) 72
    4. d) 48

  2. A committee of 2 people is chosen from a group of 10. How many different committees are possible?

    1. a) 45
    2. b) 90
    3. c) 20
    4. d) 100

  3. How many 4-digit numbers can be formed from the digits 1–9 if no digit is repeated?

    1. a) 504
    2. b) 3024
    3. c) 6561
    4. d) 36

  4. A committee of 5 is chosen from 6 men and 4 women. How many committees contain at least 3 women?

    1. a) 60
    2. b) 66
    3. c) 126
    4. d) 120

  5. A lock code uses 3 digits (0–9), and digits may be repeated. How many codes are possible?

    1. a) 999
    2. b) 30
    3. c) 720
    4. d) 1000

  6. How many terms are in the expansion of (2x − 3y)⁸?

    1. a) 7
    2. b) 9
    3. c) 8
    4. d) 16

  7. A team of 4 is chosen from 7 girls and 5 boys. How many teams have exactly 2 girls and 2 boys?

    1. a) 420
    2. b) 210
    3. c) 70
    4. d) 495

  8. How many different arrangements can be made using all the letters of the word BANANA?

    1. a) 360
    2. b) 120
    3. c) 720
    4. d) 60

  9. Which row of Pascal's triangle gives the coefficients in the expansion of (a + b)³?

    1. a) 1 4 6 4 1
    2. b) 1 3 3 1
    3. c) 1 2 1
    4. d) 1 3 1

  10. A pizza shop offers 6 toppings. How many different pizzas with exactly 3 different toppings can be made?

    1. a) 216
    2. b) 20
    3. c) 18
    4. d) 120

Answer key · Counting · Set B

Math 30-1 · maddyhelps.com

  1. c) 72 — Place the 3 consonants first: 3! = 6. That makes 4 gaps; choose 2 for the vowels and order them: ₄C₂ × 2! = 12. Then 6 × 12 = 72.
  2. a) 45 — No roles on the committee, so order does not matter: ₁₀C₂ = (10 × 9) ÷ 2 = 45. 90 is ₁₀P₂, counting "Ana and Ben" separately from "Ben and Ana".
  3. b) 3024 — Order matters and repeats are banned: ₉P₄ = 9 × 8 × 7 × 6 = 3024. 6561 is 9⁴, which would allow repeats.
  4. b) 66 — "At least 3" splits into exactly 3 and exactly 4: ₄C₃ × ₆C₂ + ₄C₄ × ₆C₁ = 4 × 15 + 1 × 6 = 66. Each case is counted separately, then added.
  5. d) 1000 — Repeats are allowed, so every position keeps all 10 digits: 10 × 10 × 10 = 1000 (the codes 000 to 999). 999 forgets 000.
  6. b) 9 — The number of terms is n + 1 = 9. The coefficients 2 and −3 change the numbers in front but never the number of terms.
  7. b) 210 — Choose each group separately, then multiply: ₇C₂ × ₅C₂ = 21 × 10 = 210. Adding instead of multiplying is the usual slip here.
  8. d) 60 — 6 letters give 6! = 720, but A repeats 3 times and N twice: 720 ÷ (3! × 2!) = 720 ÷ 12 = 60.
  9. b) 1 3 3 1 — Row 3 is 1 3 3 1, matching (a + b)³ = a³ + 3a²b + 3ab² + b³. Rows start at row 0, so 1 4 6 4 1 is row 4.
  10. b) 20 — The order toppings are added does not change the pizza: ₆C₃ = 720 ÷ (6 × 6) = 20. 120 is ₆P₃, which counts the same pizza 6 times.