Permutations and combinations
Permutations and combinations
- A permutation is an arrangement where order matters: ${}^nP_r = \dfrac{n!}{(n-r)!}$.
- A combination is a selection where order doesn't matter: ${}^nC_r = \dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$.
- To arrange a word with repeats, divide by the factorial of each repeat count.
Practice
How many ways are there to choose 2 from 5 (⁵C₂)?
⁵C₂ = 5!/(2!3!) = 120/(2×6) = 10.
Practice
How many arrangements of the letters of NEEDLESS? (8 letters: E×3, S×2 → 8!/(3!2!))
8!/(3!2!) = 40320/(6×2) = 40320/12 = 3360.
Practice
In a combination, the order of the chosen items:
Combinations ignore order; permutations count order.
You've got it
Key idea
- permutation (order matters): ${}^nP_r = \dfrac{n!}{(n-r)!}$
- combination (order doesn't): ${}^nC_r = \dfrac{n!}{r!(n-r)!}$
- repeated letters: divide $n!$ by each repeat's factorial (NEEDLESS → $\dfrac{8!}{3!\,2!}$)