Permutations and Combinations

Learning Objective: to determine the number of combinations are possible when order does or doesn’t matter

Warm-up (IN)

1. a. How many 5 letter code words are possible from the first 7 letters of the alphabet if no letter is repeated?b. If letters are repeated?c. If adjacent letters must be different?

2. How many different pizzas can you order if there are 2 types of crust, 8 toppings, 3 sauces and 2 cheeses to choose from?

Notes!

POM - Permutation Order Matters!

Permutation -

Ex 1 – How many ways can you arrange 5 books on a shelf

Factorial = ! on the calculator: Math, cursor to PRB

A permutation of n objects taken r at a time...

an arrangement of objects in a specific order without repetition

5 4 3 2 1 = 120

Shortcut!

,P n n or n nP = !n

,P n r or n rP = !

!

n

n r

Ex 2 - 5 books but only putting 3 on the shelf

Ex 3 - How many permutations of {A,B,C,D} if 2 at a time?

Ex 4 - How many permutations of 10 objects taken 4 at a time?

5 3P = 5!

5 3 !=5 4 3 2 1

2 1

= 60

4 2P = 12

10 4P = 5040

- fewer than the number of permutations because the arrangement does not matter!

Combination - a set of n objects taken r at a time without repetition

i.e. AB = BA

,C n r or n rC = !

! !

n

r n rEx 5 - How many 4-person committees can be formed from a group of 10 people? Does order matter? Y or N

Perm or Comb?

10 4C =10!

4!6!=10 9 8 7 6 5 4 3 2 1

4 3 2 1 6 5 4 3 2 1

= 210

video

Ex 6 – a. If 8 people want to hold office in an organization, how many ways to choose pres, vp, sec, and treas?

b. How many ways to get a committee of 3?

**To determine whether you need to find permutations or combinations, see if rearranging the objects produces a different result!

Does order matter? Y or N

Perm or Comb?8 4P = 1680

Does order matter? Y or N

Perm or Comb?8 3C = 56

HW –

Out – How many different ice cream cones can I get if there are 31 flavors and I want 3 scoops?

Summary – The difference between permutations and combinations is…

Don’t forget about POW!!