Permutations & Combinations

🧠

Quick recall

• Decide whether order matters, then handle restrictions first.
• Circular counts identify rotations as the same and reflections as different unless stated otherwise.

On this page

Formula Cards

1. Factorial

$$n!=n(n-1)\cdots1,\qquad0!=1$$

2. Permutation — order matters

$${}^nP_r=\frac{n!}{(n-r)!}$$

Seating, ranking, and ordered selections of distinct items without repetition.

3. Arrange all distinct items

$$n!$$

4. Circular permutation

$$(n-1)!$$

Rotations identical; reflections different.

5. Combination — selection only

$${}^nC_r=\frac{n!}{r!(n-r)!}$$

Teams and committees: selection order does not matter.

6. Combination symmetry

$${}^nC_r={}^nC_{n-r}$$

Worked Examples

Example 1

Question: Evaluate 5 factorial.

Step 1: Expand the factorial.

$$5!=5\times4\times3\times2\times1$$

Step 2: Multiply.

$$5!=120$$

Answer: 120.

Example 2

Question: Arrange 3 books chosen from 5 distinct books on a shelf.

Step 1: Order matters, so use permutation.

$${}^5P_3=\frac{5!}{(5-3)!}$$

Step 2: Evaluate.

$$\frac{5!}{2!}=5\times4\times3=60$$

Answer: 60 arrangements.

Example 3

Question: Choose 3 students out of 5 for a team.

Step 1: Order does not matter, so use combination.

$${}^5C_3=\frac{5!}{3!\,2!}$$

Step 2: Evaluate.

$${}^5C_3=10$$

Answer: 10 teams.

Example 4

Question: Arrange 4 distinct people around a round table, with rotations counted as identical.

Step 1: Fix one person's position to remove rotational duplicates.

$$(4-1)!$$

Step 2: Arrange the remaining three people.

$$3!=6$$

Answer: 6 arrangements.

Example 5

Question: From 6 men and 4 women, choose a committee of 3 men and 2 women.

Step 1: Count the selections separately.

$${}^6C_3=20,\qquad{}^4C_2=6$$

Step 2: Multiply the independent selection counts.

$$20\times6=120$$

Answer: 120 committees.

Drill

Try each question before opening its answer.

Question 1

How many ways can 4 letters A,B,C,D be arranged?

• Show answer

  24

Question 2

How many 3-letter arrangements from 6 distinct letters (permutation)?

• Show answer

  120

Question 3

Choose 4 people from a group of 7 for a committee.

• Show answer

  35

Question 4

Circular arrangement of 6 people.

• Show answer

  120

Question 5

From 5 red and 3 blue balls, choose 2 red and 1 blue.

• Show answer

  30

Permutations with Restrictions

• When objects are arranged under restrictions (specific positions, items grouped/separated, etc.), settle the restriction first, then apply the basic counting principle (multiply remaining choices) to the rest.
• Common restriction patterns:
  • Fixed first/last position: fill the restricted slot first (fewer choices), then arrange the rest normally.
  • Items must be together: glue them into one block, arrange the block + others, then arrange inside the block.
  • Items must NOT be together: arrange everything else first, then insert the restricted items into the gaps.
  • No repetition, digits from a set: once a digit is used for one position, it's removed from the pool for the next.

Restricted Digit Examples

Example 1

Question: Form five-digit numbers without repetition from 1,3,4,5,6,7,8,9 that are less than 40,000.

Step 1: Restrict the first digit to 1 or 3.

$$2\text{ choices}$$

Step 2: Fill four ordered places from the seven remaining digits.

$${}^7P_4=7\times6\times5\times4=840$$

Step 3: Multiply.

$$2\times840=1680$$

Answer: 1,680.

Example 2

Question: Using the same eight digits without repetition, form even five-digit numbers.

Step 1: Restrict the last digit to 4, 6, or 8.

$$3\text{ choices}$$

Step 2: Fill the other four places from seven remaining digits.

$${}^7P_4=840$$

Step 3: Multiply.

$$3\times840=2520$$

Answer: 2,520.

All Restriction Types — Worked Examples

1. Fixed position

Formula

$$(n-1)!$$

Question: Six people A–F in a row; A must sit first.

Step 1: Fix A in the first seat.

$$1\text{ choice}$$

Step 2: Arrange the remaining five people.

$$5!=120$$

Answer: 120.

2. One item allowed in k positions

Formula

$$k(n-1)!$$

Question: Six people in a row; A must sit first or last.

Step 1: Choose A's seat.

$$k=2$$

Step 2: Arrange the remaining five.

$$2\times5!=240$$

Answer: 240.

3. Two specific items together

Formula

$$(n-1)!\,2!$$

Question: Five distinct books; two specific books must be together.

Step 1: Treat the pair as one block.

$$\text{Units}=3+1=4$$

Step 2: Arrange units and order the pair internally.

$$4!\times2!=48$$

Answer: 48.

4. Two specific items apart

Formula

$$n!-(n-1)!\,2!$$

Question: The same five books; the pair must not be together.

Step 1: Count all arrangements.

$$5!=120$$

Step 2: Subtract together cases.

$$120-48=72$$

Answer: 72.

5. A group of k items together

Formula

$$(n-k+1)!\,k!$$

Question: Four boys and three girls in a row; all girls together.

Step 1: Treat the girls as one block.

$$\text{Units}=4+1=5$$

Step 2: Arrange units and girls within their block.

$$5!\times3!=720$$

Answer: 720.

6. No two of k restricted items adjacent

Formula

$$m!\,{}^{m+1}P_k$$

Question: Four boys and three girls in a row; no two girls together.

Step 1: Arrange the boys.

$$4!=24$$

Step 2: Count five gaps, including the ends, and place distinct girls.

$${}^5P_3=5\times4\times3=60$$

Step 3: Multiply.

$$24\times60=1440$$

Answer: 1,440. Requires k ≤ m+1; here m = 4, k = 3.

7. Circular arrangement with a pair together

Formula

$$(n-2)!\,2!$$

Question: Six people around a round table; two specific people adjacent.

Step 1: Treat the pair as a block: five circular units.

$$(5-1)!=24$$

Step 2: Count the pair's two internal orders.

$$24\times2=48$$

Answer: 48; rotations identical, reflections different.

8. Restricted first position from a symbol pool

Formula

$$(n-j)\,{}^{n-1}P_{r-1}$$

Question: Make four-digit numbers without repetition from 0,1,2,3,4,5; first digit cannot be 0.

Step 1: Choose the first digit from five nonzero digits.

$$6-1=5$$

Step 2: Fill three positions from five remaining digits.

$${}^5P_3=60$$

Step 3: Multiply.

$$5\times60=300$$

Answer: 300. Formula assumes distinct symbols, no repeats, and only the first-position restriction.

9. One item before another

Formula

$$\frac{n!}{2}$$

Question: Arrange A,B,C,D,E in a row with A before B; adjacency is not required.

Step 1: Count unrestricted arrangements.

$$5!=120$$

Step 2: Half have A before B.

$$\frac{120}{2}=60$$

Answer: 60.

10. Alternating equal groups

Formula

$$2(k!)^2$$

Question: Three boys and three girls in a row, alternating.

Step 1: Choose one of the two patterns.

$$BGBGBG\quad\text{or}\quad GBGBGB$$

Step 2: Arrange each group within its slots.

$$3!\times3!=36$$

Step 3: Multiply by two patterns.

$$2\times36=72$$

Answer: 72. Formula assumes distinct people and equal group sizes.

Restriction Drill

Try each question before opening its answer.

Question 1

How many 4-digit numbers (no repetition) from 1,2,3,4,5,6 are less than 4000?

• Show answer

  (3 choices for 1st) × ⁵P₃ = 3×60 = 180

Question 2

How many 4-digit numbers (no repetition) from 1,2,3,4,5,6 are odd?

• Show answer

  (3 choices for last: 1,3,5) × ⁵P₃ = 3×60 = 180

Question 3

In how many ways can 5 boys and 3 girls sit in a row if the girls must sit together?

• Show answer

  treat girls as 1 block → $6!\times3!=4320$

Question 4

In how many ways can 6 people be arranged in a row if 2 specific people must NOT sit together?

• Show answer

  total $6!=720$ minus together $5!\times2!=240$ → 480

Practice tracker

YYYY-MM-DD

• Attempted/Correct:
• Mistake pattern: