1. Section 3 – Permutations & Combinations

2. Learning Objectives Differentiate between: – Permutations: Ordering “r” of “n” objects (no replacement) Special case: ordering “n” of “n” objects – Combinations: Selecting without order, “r” of “n” objects (no replacement) Binomial Theorem Multinomial Theorem

3. Permutations Permute: ordered (no replacement) Permuting “r” of “n” objects: Special case: permuting all n distinct objects (n=r)

4. Combinations Combining: unordered (no replacement) Combining “r” of “n” objects: – Called “n choose r”

5. Comparing Combinations & Permutations Combinations has an r! term in the denominator: so why are there less combinations than permutations? – Ex: Consider set {a, b, c} & we want to choose 2 Permutations: {a, b} {b, a} {a, c} {c, a} {b, c} {c, b} – Order matters! Combinations: {a, b} {b, c} {c, a} – Order does NOT matter!

6. Binomial Theorem Combinations are used in the power series expansion of (1+t)^N to find the coefficient of each term This will be useful for binomial distributions later (don’t worry about memorizing it now, but make sure you understand it when we get to the binomial distribution)

7. Multinomial Theorem Given n objects, (n1 of type 1, n2 of type 2, …ns of type s) choose k1 of type 1, etc… Used in multinomial distribution later

8. Summary Order matters? Order doesn’t matter?