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Math 30-1 · Worksheets

Counting · Set A

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 A

Math 30-1 · maddyhelps.com

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

  1. How many terms are in the expansion of (x + y)⁶?

    1. a) 5
    2. b) 12
    3. c) 7
    4. d) 6

  2. How many different 5-card hands can be dealt from a standard 52-card deck?

    1. a) 22 100
    2. b) 311 875 200
    3. c) 270 725
    4. d) 2 598 960

  3. Solve for n: ₙP₂ = 42

    1. a) 8
    2. b) 7
    3. c) 6
    4. d) 21

  4. In how many ways can 5 people sit in a row if two particular people must sit together?

    1. a) 24
    2. b) 96
    3. c) 48
    4. d) 120

  5. What is the constant term in the expansion of (x + 1/x)⁶?

    1. a) 15
    2. b) 20
    3. c) 6
    4. d) 64

  6. In the expansion of (2x − y)⁵, what is the coefficient of x²y³?

    1. a) 80
    2. b) −40
    3. c) −80
    4. d) 40

  7. What is the value of 5! ?

    1. a) 720
    2. b) 60
    3. c) 25
    4. d) 120

  8. In how many ways can 4 different books be arranged in a row on a shelf?

    1. a) 16
    2. b) 4
    3. c) 256
    4. d) 24

  9. Evaluate ₈P₂.

    1. a) 64
    2. b) 16
    3. c) 56
    4. d) 28

  10. A committee of 3 is chosen from 8 people, and one of the three is then named chair. How many outcomes are possible?

    1. a) 56
    2. b) 168
    3. c) 336
    4. d) 512

Answer key · Counting · Set A

Math 30-1 · maddyhelps.com

  1. c) 7 — An expansion of (a + b)ⁿ has n + 1 terms, so 6 + 1 = 7. Counting k from 0 to 6 gives 7 values, not 6.
  2. d) 2 598 960 — A hand is a group, not an order: ₅₂C₅ = 2 598 960. The larger number 311 875 200 is ₅₂P₅, which counts the same hand in every dealing order.
  3. b) 7 — ₙP₂ = n(n − 1), so n(n − 1) = 42. Two consecutive numbers multiplying to 42 are 7 and 6, so n = 7.
  4. c) 48 — Glue the pair into one block, leaving 4 items to arrange: 4! = 24. The pair can sit in 2 orders inside the block, so 24 × 2 = 48.
  5. b) 20 — The general term gives x⁶⁻ᵏ × x⁻ᵏ = x⁶⁻²ᵏ. A constant needs 6 − 2k = 0, so k = 3 and the term is ₆C₃ = 20.
  6. b) −40 — Take k = 3: ₅C₃ (2x)² (−y)³ = 10 × 4x² × (−y³) = −40x²y³. The minus sign survives because it is raised to an odd power.
  7. d) 120 — 5! = 5 × 4 × 3 × 2 × 1 = 120. It is not 5 × 5, and it does not stop partway — keep multiplying all the way down to 1.
  8. d) 24 — All 4 books are used and order matters, so it is 4! = 4 × 3 × 2 × 1 = 24. 256 is 4⁴, which would let one book sit in more than one spot.
  9. c) 56 — ₙPᵣ is r factors counting down from n, so ₈P₂ = 8 × 7 = 56. 28 is ₈C₂, which ignores order.
  10. b) 168 — Choose the group, then pick the chair: ₈C₃ × 3 = 56 × 3 = 168. You can also pick the chair first: 8 × ₇C₂ = 8 × 21 = 168.