ACT · Math · Preparing for Higher Math: Statistics and Probability

Counting and combinatorics

Counting and combinatorics is part of Preparing for Higher Math: Statistics and Probability in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include multiplication principle, arrangements of distinct items, selection where order does not matter and arrangements with repeated items.

Common traps: adding the choices; multiplying only some of the stages; counting a choice that is not allowed.

Sample questions

Question 1
A community kitchen needs 3 volunteers for Saturday, and 10 people have offered to help. In how many ways can the group of 3 volunteers be chosen?
- A.
  120
- B.
  360
- C.
  720
- D.
  1,000
Answer: A. 120
The group has no roles, so order does not matter: C(10, 3) = (10 × 9 × 8)/(3 × 2 × 1) = 120. Choice B divides by 2 instead of 3!; C counts ordered selections, 10 × 9 × 8; D is 10³, allowing the same person more than once.
Question 2
In a race with 9 runners, how many different ways can 1st through 4th place be awarded, assuming no ties?
- A.
  36
- B.
  126
- C.
  3024
- D.
  6561
Answer: C. 3024
Order matters, so use permutations: 9 × 8 × 7 × 6 = 3024.
Question 3
How many diagonals does a convex polygon with 20 sides have?
- A.
  170
- B.
  190
- C.
  340
- D.
  380
Answer: A. 170
Each vertex joins to the 17 vertices that are not itself or its neighbors, and each diagonal is counted from both ends: 20 × 17 ÷ 2 = 170. Choice B counts all segments between vertices, including the 20 sides; C does not divide by 2; D is 20 × 19 without dividing or removing sides.
Question 4
A 4-digit door code uses four different digits. The first digit cannot be 0, and the last digit must be odd. How many codes are possible?
- A.
  2,240
- B.
  2,520
- C.
  4,536
- D.
  5,040
Answer: A. 2,240
Fill the restricted places first. Last digit: 5 odd choices. First digit: not 0 and not the last digit, 8 choices. Middle digits: 8 × 7. Total 5 × 8 × 8 × 7 = 2,240. Choice B ignores the no-zero rule for the first digit; C ignores the odd rule (9 × 9 × 8 × 7); D ignores both rules (10 × 9 × 8 × 7).
Question 5
A pizzeria offers 8 different toppings. Every pizza must have at least 2 different toppings, and no topping may be used twice. How many different topping combinations are possible?
- A.
  247
- B.
  248
- C.
  255
- D.
  256
Answer: A. 247
There are 2⁸ = 256 subsets of the toppings. Remove the empty set and the 8 single-topping sets: 256 − 1 − 8 = 247. Choice B removes the single-topping sets but not the empty one; C removes only the empty set; D removes nothing.

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