Section 3 – Permutations & Combinations

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

Permutations

• Permute: ordered (no replacement)• Permuting “r” of “n” objects:

nPr n!

(n r)!

• Special case: permuting all n distinct objects (n=r)

nPn n!

(n n)!n!

(0)!n!

1n!

Combinations

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

n

r

nCr

n!

r!(n r)!

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!

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)

(1 t)N N

k

t k 1Nt

N(N 1)2k0

t 2 ...

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

N

k1k2...ks

N!

k1!k2!... ks!

Summary

• Order matters?

• Order doesn’t matter?

nPr n!

(n r)!

n

r

nCr

n!

r!(n r)!