Permutations & Combinations

Scenario

Picking a 4-digit UPI PIN (where order matters — 1234 is different from 4321) vs picking 3 friends for a Goa trip (where Alice-Bob-Carol is the same group as Carol-Bob-Alice).

How do you count the right number of possibilities without listing them all?

Mental model

Permutation = arrange (order matters). Combination = choose (order doesn't).

If swapping two items creates a new valid outcome (password 123 vs 321), it's a Permutation. If swapping changes nothing (team of Alice+Bob is the same team both ways), it's a Combination. nCr = nPr / r! because combinations kill the r! duplicate orderings.

Formula

nPr = n! / (n-r)! (arrange r from n). nCr = n! / (r! x (n-r)!) (choose r from n). Repeated items: arrange n items where one group of p is identical, another q is identical -> n! / (p! x q!). Circular arrangements: (n-1)! for distinct items around a circle.

nCr = nC(n-r) by symmetry — choosing r to include is the same as choosing (n-r) to exclude. For repeated items, divide by the factorial of each repeated group's size.

Worked example

Problem: How many distinct arrangements can be made from the letters of the word GOOGLE?

1. Set up: GOOGLE has 6 letters. Check for repeats: G appears 2 times, O appears 2 times, L and E appear once each.
2. Formula: arrangements with repeats = n! / (p! x q! x ...) = 6! / (2! x 2!) (for the two G's and two O's).
3. Compute: 6! = 720. 2! x 2! = 2 x 2 = 4. Total = 720 / 4 = 180.
4. Sanity-check: Two repeated groups each halve the count. Starting from 720, halve twice gives 720/4 = 180. Makes sense.

Answer: 180 distinct arrangements.

Problem: A team of 4 is to be formed from 5 men and 4 women such that there is exactly 1 woman. How many ways?

1. Set up: Need 1 woman AND 3 men. Order within the team doesn't matter (team, not lineup).
2. Count women choices: 4C1 = 4.
3. Count men choices: 5C3 = 5! / (3! x 2!) = (5 x 4) / 2 = 10.
4. Combine (AND = multiply): 4C1 x 5C3 = 4 x 10 = 40.
5. Sanity-check: Total ways to pick 4 from 9 = 9C4 = 126. 40 out of 126 is plausible for the 1-woman restriction.

Answer: 40 ways.

Shortcuts

• nCr Complement Rule: nCr = nC(n-r). If r > n/2, flip it to make calculation easier. (100C98 = 100C2 = (100 x 99) / 2 = 4950. Done in 3 seconds.)
• AND vs OR in Counting: 'AND' (both conditions true) = multiply the counts. 'OR' (either condition) = add the counts. (3 shirt choices AND 4 pants choices = 3 x 4 = 12 outfits. Red shirt OR blue shirt = 1 + 1 = 2 choices.)

Common mistakes

• Using Permutation instead of Combination: Selecting a committee of 3 from 10 people requires 10C3. Many mistakenly use 10P3, which assumes the 3 people have distinct roles.
• Forgetting to divide for repeated items: MOON has 2 O's. Arrangements = 4!/2! = 12, not 4! = 24. Every pair of arrangements that only differ in swapping the two O's is actually one arrangement.

Glossary

symmetry

When two things perfectly balance or mirror each other.

factorial

Multiplying a number by every whole number smaller than it down to 1.

outcomes

The final results or possibilities from a situation.

Recall questions

• What is the formula for nCr?
• How many distinct arrangements of the letters in MOON?
• Is selecting a committee a permutation or combination problem?

Questions & answers

How many words can be formed using the letters of the word 'AMAZON'?

360

Approach: 6 letters, A repeats 2 times. Arrangements = 6! / 2! = 720 / 2 = 360.

A team of 4 is to be formed from 5 men and 4 women such that there is exactly 1 woman. How many ways?

40

Approach: 1 woman from 4 (4C1 = 4) AND 3 men from 5 (5C3 = 10). Multiply: 4 x 10 = 40.

Continue learning

Next: Probability

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