Permutations & Combinations

Every question in this topic starts with one decision: does the order matter?

• Yes → permutation. Arranging, seating, forming numbers, ranking. nPr = n!/(n−r)!
• No → combination. Selecting, choosing a committee, drawing balls. nCr = n!/(r!(n−r)!)

A one-second test: swap two of the chosen items. If that counts as a different answer, it is a permutation.

Four standard patterns:

1. Arrange n distinct things → n!.
2. Arrange a word with repeated letters → n! divided by the factorial of each repeat count. LEADER has 6 letters with E twice → 6!/2!.
3. "All the vowels together" → tie the vowels into one block. Arrange the block with the consonants, then arrange the vowels inside the block, and multiply the two.
4. A committee with a condition ("exactly 2 women") → choose the women and the men separately with nCr, then multiply.

The single most common mistake is adding when the rule says multiply. Separate choices that must all happen combine by multiplication; genuinely alternative cases ("2 women or 3 women") combine by addition. Ask whether the two steps happen together or instead of each other.

Useful shortcuts: nCr = nC(n−r), so 10C8 = 10C2 = 45; nC0 = nCn = 1; 0! = 1.

order matters → nPr = n!/(n−r)! · order does not → nCr = n!/(r!(n−r)!) · repeats → n!/(p!q!) · "together" → tie into a block · and = ×, or = +

In how many ways can the letters of the word LEADER be arranged? → 6 letters, but E appears twice, so the count is 6!/2! = 720/2 = 360. Sanity check: if all six letters were different there would be 720 arrangements; because the two E's can be swapped without producing anything new, exactly half of those 720 are duplicates — 360 genuine arrangements.