Counting problems ask “how many ways?” How many passwords, how many teams, how many orders of finishing a race. The answer comes down to one question you ask every time: does the order matter? If it does, it is a permutation. If it doesn’t, it is a combination.
The fundamental counting principle
If one choice can be made in m ways and a second in n ways, both can be made in m × n ways. This extends to any number of choices.
Example A restaurant offers 4 starters, 6 mains and 3 desserts. How many three-course meals are possible?
4 × 6 × 3 = 72.
Example How many 4-digit PINs are possible if digits can repeat?
Each of the 4 positions has 10 options: 10 × 10 × 10 × 10 = 10,000.
Example How many 4-digit PINs have no repeated digit?
10 options, then 9, then 8, then 7: 10 × 9 × 8 × 7 = 5,040.
Draw a blank for each position, write the number of options in each, and multiply. That method alone solves most counting problems on the SAT and ACT.
Factorials
n! (“n factorial”) is n × (n − 1) × … × 2 × 1. It counts the ways to arrange n different things in a row.
- 3! = 6
- 5! = 120
- 10! = 3,628,800
- 0! = 1 by definition (there is exactly one way to arrange nothing).
Example In how many orders can 6 runners finish a race (no ties)?
6! = 720.
Permutations: order matters
A permutation is an arrangement where order matters. The number of ways to pick and arrange r things from n is:
nPr = n!/(n − r)!
Example A club of 12 people elects a president, vice president and treasurer. How many outcomes are possible?
- Order matters, because being president differs from being treasurer.
- 12P3 = 12!/9! = 12 × 11 × 10 = 1,320.
The blank-and-multiply method gives the same thing: 12 choices for president, then 11, then 10.
Combinations: order doesn’t matter
A combination is a selection where order doesn’t matter. Choosing Ana, Ben and Cal for a team is the same team as Cal, Ana and Ben. The formula is:
nCr = n!/(r!(n − r)!)
It is also written as (n choose r). It equals the permutation count divided by r!, because each group of r can be arranged r! ways and we only want to count it once.
Example A club of 12 people chooses a 3-person committee. How many committees are possible?
- Order doesn’t matter: there are no roles.
- 12C3 = 1,320/3! = 1,320/6 = 220.
Quick test: swap two people
If swapping two chosen people gives a different outcome, order matters (permutation). If it gives the same outcome, it doesn’t (combination).
Side by side
| Question | Order matters? | Count |
|---|---|---|
| Top 3 finishers (1st, 2nd, 3rd) of 8 runners | yes | 8P3 = 336 |
| 3 runners of 8 who qualify for the final | no | 8C3 = 56 |
| 5-card hands from a 52-card deck | no | 52C5 = 2,598,960 |
| 4-letter codes from 26 letters, no repeats | yes | 26P4 = 358,800 |
Arrangements with repeated items
When some items are identical, divide by the factorial of each repeat count, because swapping identical items doesn’t create a new arrangement.
Example How many distinct arrangements of the letters in BANANA are there?
- 6 letters: B ×1, A ×3, N ×2.
- 6!/(3! × 2!) = 720/12 = 60.
Combinations with conditions
Break the selection into parts and multiply.
Example A committee of 4 is chosen from 7 juniors and 5 seniors. How many committees have exactly 2 seniors?
- Choose 2 of 5 seniors: 5C2 = 10.
- Choose 2 of 7 juniors: 7C2 = 21.
- Multiply: 10 × 21 = 210.
For “at least one” questions, it is usually easier to count the total and subtract the cases with none.
Example How many 4-person committees from the same group have at least one senior?
- All committees: 12C4 = 495.
- Committees with no seniors (all juniors): 7C4 = 35.
- At least one senior: 495 − 35 = 460.
Probability with combinations
Probability is favorable outcomes over total outcomes, and combinations count both. The probability basics guide covers the rules this builds on.
Example A bag has 5 red and 4 blue marbles. Three are drawn at random. What is the probability all three are red?
- Ways to choose 3 red: 5C3 = 10.
- Ways to choose any 3: 9C3 = 84.
- Probability: 10/84 = 5/42 ≈ 0.119.
On a calculator, nPr and nCr are under MATH → PRB on a TI-84. In Desmos, type nPr(12,3) or nCr(12,3).
Your turn
How many license plates have 3 letters followed by 3 digits, with repeats allowed?
Show answer
17,576,000. 26³ × 10³ = 17,576 × 1,000.
In how many ways can 5 books be arranged on a shelf?
Show answer
120. 5! = 120.
How many ways can a coach choose 5 starters from a team of 11?
Show answer
462. Order doesn’t matter: 11C5 = 462.
How many ways can gold, silver and bronze be awarded among 10 finalists?
Show answer
720. Order matters: 10 × 9 × 8.
How many distinct arrangements of the letters in LETTER are there?
Show answer
180. 6 letters with E ×2 and T ×2: 720/(2 × 2).
Two cards are drawn from a standard deck without replacement. What is the probability both are hearts?
Show answer
1/17. 13C2/52C2 = 78/1,326 = 1/17.