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:
n!
n Pr 
(n  r)!
• Special case: permuting all n distinct objects (n=r)

n!
n! n!

  n!
n Pn 
(n  n)! (0)! 1
Combinations
• Combining: unordered (no replacement)
• Combining “r” of “n” objects:
– Called “n choose r”
n
n!
 n Cr 
r!(n  r)!
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
N  k
N(N 1) 2
(1 t)    t  1Nt 
t  ...
k
2
k 0  

N
• 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)
Multinomial Theorem
• Given n objects, (n1 of type 1, n2 of type 2,
…ns of type s) choose k1 of type 1, etc…
 N 
N!


k1k2 ...ks  k1!k2!... ks!
• Used in multinomial distribution later

Summary
n!
n Pr 
(n  r)!
• Order matters?
• Order doesn’t matter?


n
n!
 n Cr 
r!(n  r)!
r 