Mathematics · Combinatorics

Counting &
Combinatorics

Exam-Style Practice — 20 Short-Answer Problems

High School / University Entrance Show Your Work 5 Topic Areas 20 Questions
Section 01

Concepts & Key Formulas

Study each topic, memorize the formulas, then work through the example before moving on.

1. Fundamental Counting Principle
  • If event A can occur in m ways and event B in n ways, then A and B together = m × n ways. (Product Rule)
  • If A can occur in m ways and B in n ways (mutually exclusive), then A or B = m + n ways. (Sum Rule)
Product Rule: m × n Sum Rule: m + n
Example: A restaurant offers 3 soups and 5 main courses. How many different meals (1 soup + 1 main) are possible?
→ 3 × 5 = 15 meals
2. Permutations
  • A permutation is an arrangement where ORDER matters.
  • P(n, r) = n! / (n−r)! — arrange r from n distinct objects.
  • Circular permutation of n objects = (n−1)!
  • With repeated elements: if groups of p, q, r are identical → n! / (p! q! r!)
nPr = n! / (n−r)! Circular: (n−1)! Repeated: n! / (p!q!r!)
Example: How many ways can 5 students be arranged in a line?
→ 5! = 120
3. Combinations
  • A combination is a selection where ORDER does NOT matter.
  • C(n, r) = n! / (r!(n−r)!) — also written nCr.
  • Symmetry identity: C(n, r) = C(n, n−r)
  • Pascal's identity: C(n, r) = C(n−1, r−1) + C(n−1, r)
nCr = n! / (r!(n−r)!) C(n,r) = C(n, n−r) Pascal: C(n,r) = C(n−1,r−1)+C(n−1,r)
Example: How many ways can 3 students be selected from a group of 8?
→ C(8,3) = 8! / (3!·5!) = 56
4. Permutations vs. Combinations — Decision Rule
  • Ask: Does order matter?
  • YES → Use Permutation (P)
  • NO → Use Combination (C)
  • Relationship: P(n,r) = C(n,r) × r!
nPr = nCr × r!
Example: Choosing president & VP from 10 people (order matters) vs. choosing a committee of 2 (order doesn't matter).
→ P(10,2) = 90 | C(10,2) = 45
5. Combinations with Repetition (Stars & Bars)
  • Choosing r items from n types, repetition allowed, order irrelevant:
  • H(n, r) = C(n+r−1, r)
  • Equivalent to distributing r identical objects into n distinct bins.
  • Also used to count non-negative integer solutions to x₁ + x₂ + … + xₙ = r.
H(n,r) = C(n+r−1, r)
Example: Choose 3 fruits from 4 types (repetition allowed).
→ H(4,3) = C(6,3) = 20
Section 02

Exam Problems

Write your answer in each field. Show your full reasoning on paper. Answers are in Section 03.

Problem 1 Fundamental Counting Principle
A password consists of 2 distinct letters (from A–Z) followed by 3 distinct digits (from 0–9). How many different passwords can be formed?
Answer
Problem 2 Sum Rule
A student must choose one elective from either the Science department (4 courses) or the Arts department (6 courses). How many choices does the student have?
Answer
Problem 3 Basic Permutation
Eight runners compete in a race. In how many ways can the gold, silver, and bronze medals be awarded?
Answer
Problem 4 Permutation with Condition
How many 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5 (each exactly once) such that the number is even?
Answer
Problem 5 Circular Permutation
Six people sit around a circular table. In how many distinct ways can they be seated if two specific people, A and B, must always sit next to each other?
Answer
Problem 6 Permutation with Repetition
How many distinct arrangements are there of the letters in the word MISSISSIPPI?
Answer
Problem 7 Combination — Complementary
A committee of 4 is chosen from 10 people (6 men and 4 women). How many committees contain at least 1 woman?
Answer
Problem 8 Combination — Two Groups
From 5 math books and 4 science books, how many ways can you choose 3 math books and 2 science books?
Answer
Problem 9 Combination Identity
Find the value of n if C(n, 2) = 45.
Answer
Problem 10 Combination with Restriction
A team of 5 is selected from 8 players. Player A refuses to play if Player B is on the team. How many valid teams are there?
Answer
Problem 11 Combinations with Repetition
How many non-negative integer solutions does the equation x + y + z = 10 have?
Answer
Problem 12 Combinations with Repetition — Bounded
How many non-negative integer solutions does x + y + z = 10 have where x ≥ 2, y ≥ 1, z ≥ 0?
Answer
Problem 13 Mixed — Permutation & Combination
A club has 12 members. They elect a president and a secretary, then choose a 3-person social committee (officers cannot be on the committee). In how many ways can this be done?
Answer
Problem 14 Grid Paths
On a 5×4 grid, how many shortest paths exist from the bottom-left corner to the top-right corner, moving only right or up?
Answer
Problem 15 Binomial Coefficient Sum
Compute: C(8,0) + C(8,1) + C(8,2) + … + C(8,8).
Answer
Problem 16 Permutation — No Two Adjacent
5 boys and 3 girls stand in a line. In how many ways can they be arranged so that no two girls are adjacent?
Answer
Problem 17 Subset Counting
How many 3-element subsets of {1, 2, 3, 4, 5, 6, 7} contain the element 4?
Answer
Problem 18 Derangements
Six cards labeled 1–6 are given to 6 players. In how many ways can the cards be distributed so that no player receives the card matching their own number?
Answer
Problem 19 Dividing into Groups
In how many ways can 9 students be divided into 3 unlabeled groups of 3?
Answer
Problem 20 Inclusion-Exclusion
How many integers from 1 to 100 are divisible by 2, 3, or 5?
Answer
0 / 20 answered
Section 03

Answer Key & Full Solutions

Click any problem to expand the full solution. Focus on the method, not just the final answer.