Permutations and

Combinations

Adapted from resources at:http://teachers.henrico.k12.va.us/math/HCPSAlgebra2/12-

2.htmland

http://www.mathsisfun.com/combinatorics/combinations-permutations.html

http://calculator.maconstate.edu/permutation/index.html

AII.12 2009

What do you call this thing?

It’s a combination lock, right?

Permutations & Combinations

A permutation is an arrangement of items in a particular order.

Notice, ORDER MATTERS!

A combination is an arrangement of items in which ORDER DOES NOT MATTER.

What should we call this thing?

So it’s really a permutation lock.

How many ways can we arrange the letters ABC in groups of 2?

____ ____

How many choices are there for first blank? 3 ____

3 2 For the second blank?

(3)(2) = 6

AB AC BA BC CA CB

How could we do it if we didn’t want to write them all out?

Is this a permutation or a combination?

It’s a permutation, because order matters.(For example, BA and AB are different things.)

PermutationsTo find the number of Permutations of n items chosen r at a time, you can use the formula !

where 0 .( )!

nP r nn r n r

The exclamation point represents a factorial.

n! is the product of all whole numbers less than or equal to n.

For example, 5! = (5)(4)(3)(2)(1) 0! is defined to be 1.

3 2

3!

(3 2)!P

3!

1!

3 2 1

1

6

Permutations

For the ABC example, we had 3 letters, and chose 2:

In the TI-83…

10 7P

If you wanted to do a permutation of 10 objects in groups of 7:

Permutations

A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

Practice:

30 3P ¿24 ,360

Combinations

To find the number of combinations of n items chosen r at a time, you can use the formula

. 0 where nrrnr

nrCn

)!(!

!

Or, on the calculator, you’ll find the combination choice just below the permutation choice.

Combinations

For example, to choose 2 letters out of 3 in ABC, we could have:

3 2C

A & B A & C B & C

3!

2!(3 2)!

3 2 1

2 1 1

6

2

3

CombinationsTo play a particular card game, each player is dealt five cards from a standard 52-card deck. How many different hands are possible?

Practice:

52 5C 2,598,960Why is this a combination, and not a permutation?

The order doesn’t matter. (Having 4 aces and a king is thesame as getting an ace, a king, then 3 aces.)

Permutation or Combination?

From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?

24 5P

permutation

5,100,480

A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?

5 3C

Permutation or Combination?

combination

10

A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards?

2 1CCenter:

5 2CForwards:

4 2CGuards:

Thus, the number of ways to select the starting line up is (2)(10)(6) = 120.

2 1 5 2 4 2 C C C

Permutations or Combinations?

combinations