These 25 problems practice the counting principle, permutations, combinations, probability with counting, the general multiplication rule and expected value (S-CP.B.8, S-CP.B.9, S-MD.A.1, S-MD.A.2). They move from basic counts to multi-step word problems.
Write each count as a product before evaluating, and keep probabilities as exact fractions.
Problem 1: Fundamental counting principle
Find the number of possibilities.
- Ethan can choose from 4 shirts, 3 pairs of pants and 2 pairs of shoes. How many outfits are possible?
- A sandwich shop offers 5 breads, 3 meats and 4 toppings, and a sandwich has one of each. How many sandwiches are possible?
Problem 2: Factorials
Evaluate each expression.
- \(6!\)
- \(\frac{7!}{5!}\)
- \(\frac{8!}{3!\,5!}\)
Problem 3: PIN codes
A phone PIN has 4 digits chosen from 0 through 9, as in the boxes below.
- How many PINs are possible if digits may repeat?
- How many PINs are possible if no digit may repeat?
Problem 4: Permutations
Evaluate each permutation.
- \(_{8}P_{3}\)
- \(_{10}P_{2}\)
Problem 5: Combinations
Evaluate each combination.
- \(_{10}C_{4}\)
- \(_{12}C_{10}\)
Problem 6: Arranging letters
Find the number of distinct arrangements of all the letters in each word.
- MATH
- BANANA
- STATISTICS
Problem 7: Order or no order
Decide whether each situation is a permutation or a combination, then calculate.
- Choose 3 of 12 students to attend a conference.
- Award gold, silver and bronze medals among 12 runners.
Problem 8: Handshakes
Six friends meet and each person shakes hands once with every other person.
How many handshakes happen?
Problem 9: Probability with two dice
Two fair dice are rolled. The grid shows all sums.
- Find the probability that the sum is 8.
- Find the probability that the sum is at least 10.
Problem 10: Probability with cards
Two cards are dealt from a standard 52-card deck.
- Find the probability that both are hearts.
- Find the probability that both are face cards (jack, queen or king).
Problem 11: Choosing a committee
A class has 10 girls and 8 boys. Four students are chosen at random for a committee. Find the probability that all four are girls.
Problem 12: Drawing without replacement
Two marbles are drawn, one after the other and without replacement, from the bag below.
- Find the probability that both are red.
- Find the probability that the first is red and the second is blue.
Problem 13: Using the multiplication rule
Use the general multiplication rule.
- \(P(A)=0.3\) and \(P(B\mid A)=0.6\). Find \(P(A\text{ and }B)\).
- \(P(A\text{ and }B)=0.24\) and \(P(A)=0.5\). Find \(P(B\mid A)\).
Problem 14: At least one
Use the complement to find each probability.
- A fair coin is flipped 3 times . Find the probability of at least one head.
- A fair die is rolled 3 times . Find the probability of at least one six.
Problem 15: Probability distribution
Let \(X\) be the number of heads when a fair coin is flipped 3 times . Complete the table, then find \(E(X)\).
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | ? | ? | ? | ? |
Problem 16: Missing probability
A random variable \(X\) has the distribution below.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | 0.1 | 0.3 | p | 0.2 |
- Find \(p\).
- Find \(E(X)\).
- Find \(P(X\ge 2)\).
Problem 17: Free throws
Priya makes 80% of her free throws. She shoots two free throws, and the shots are independent.
- Find the probability that she makes both.
- Find the probability that she makes exactly one.
- Find the expected number of shots made.
Problem 18: Books on a shelf
Noah arranges 5 different books on a shelf.
- How many arrangements are possible?
- How many arrangements keep two particular books next to each other?
Problem 19: Student council
A school has 6 girls and 7 boys running for a 5-person committee.
- How many committees have exactly 2 girls and 3 boys?
- If the committee is chosen at random from all 13 students, find the probability it has exactly 2 girls and 3 boys.
Problem 20: License plates
A state plate has 3 letters followed by 3 digits.
- How many plates are possible if letters and digits may repeat?
- How many are possible if no letter and no digit may repeat?
Problem 21: Raffle
A school raffle sells 200 tickets at $5 each. The prizes are one $300 prize, three $50 prizes and six $10 prizes. Find the expected net result for a person who buys one ticket.
Problem 22: Insurance
Aisha’s company charges a $250 yearly premium for a policy. There is a 2% chance a customer files a $5,000 claim. Find the company’s expected profit per policy.
Problem 23: Five hearts
A 5-card hand is dealt from a standard deck. Find the probability that all five cards are hearts, as a fraction in lowest terms and as a decimal rounded to 6 places.
Problem 24: Two-stage tree
Liam draws two marbles without replacement from a jar with 4 green and 6 yellow marbles. Find the probability that the two marbles are the same color.
Problem 25: A dice game
A carnival game costs $2.50 to play. Sofia rolls one fair die and wins $12 for a six, $3 for a four or five, and nothing otherwise. Find the expected net result and say whether the game favors the player.
See every chapter on the Precalculus math page.