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Math 30-1 · Worksheets

Mixed review · Counting, exponents and logs

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

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Mixed review · Counting, exponents and logs

Math 30-1 · maddyhelps.com

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

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

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

  2. Evaluate log₂ 12 − log₂ 3.

    1. a) 4
    2. b) log₂ 9
    3. c) 3
    4. d) 2

  3. Which situation should be solved using a combination instead of a permutation?

    1. a) Arranging 3 photos in a row on a wall
    2. b) Creating a 3-letter password
    3. c) Awarding gold, silver and bronze medals to 3 of 10 runners
    4. d) Choosing 3 students from a class of 25 to attend a field trip

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

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

  5. Evaluate 2 log 5 + log 4.

    1. a) 100
    2. b) log 14
    3. c) 2
    4. d) 10

  6. If log 2 = a and log 3 = b, which expression equals log 12?

    1. a) 2a + b
    2. b) a²b
    3. c) 2ab
    4. d) a + b

  7. How many different arrangements can be made using all the letters of the word BOOK?

    1. a) 4
    2. b) 12
    3. c) 24
    4. d) 6

  8. What is the first term in the expansion of (2x + 3)³?

    1. a) 2x³
    2. b) 27
    3. c) 6x³
    4. d) 8x³

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

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

  10. Solve for n: ₙP₂ = 42

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

Answer key · Mixed review · Counting, exponents and logs

Math 30-1 · maddyhelps.com

  1. c) 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.
  2. d) 2 — Subtracting logs with the same base means dividing inside: log₂(12 ÷ 3) = log₂ 4 = 2. log₂ 9 wrongly subtracts the insides.
  3. d) Choosing 3 students from a class of 25 to attend a field trip — The same 3 students go on the trip no matter who is picked first, so order does not matter. Medals, passwords and rows all change meaning when you reorder them.
  4. d) 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.
  5. c) 2 — The power law moves the 2 inside: log 25 + log 4. The product law combines them: log 100 = 2.
  6. a) 2a + b — 12 = 2² × 3, so log 12 = log 2² + log 3 = 2a + b. Logs turn multiplying into adding and powers into multipliers.
  7. b) 12 — 4 letters give 4! = 24, but the two O's are identical, so every word was counted 2! times: 24 ÷ 2 = 12.
  8. d) 8x³ — The first term takes 2x from every bracket: (2x)³ = 2³x³ = 8x³. Writing 2x³ cubes only the x — bracket the whole 2x first.
  9. b) 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.
  10. b) 7 — ₙP₂ = n(n − 1), so n(n − 1) = 42. Two consecutive numbers multiplying to 42 are 7 and 6, so n = 7.