UNIT: PROBABILITY6-7: PERMUTATIONS AND COMBINATIONSEssential Question: How is a combination different from a permutation?

6-7: COMBINATIONS AND PERMUTATIONS

Permutation: an arrangement of items in a particular order

When all items of a particular set are used, you can alternatively use factorial notation. Example:

In how many different ways can ten dogs line up to be groomed?

Answer: There are 10 dogs to choose from first, multiplied by 9

dogs remaining to choose second, followed by 8 dogs to choose from to go third, etc.

10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 10! = 3,628,800 ways

Your turn: In how many ways can you line up 6 trophies on a shelf?

720 ways

6-7: COMBINATIONS AND PERMUTATIONS Sometimes, not all items will be used. In this

case, we can use the formula for permutations.

Example: Seven yachts enter a race. First, second, and third places

will be given. How many arrangements of 1st, 2nd, and 3rd places are possible for the seven yachts?

Answer: 7 possible 1st place finishers, 6 remaining 2nd place

finishers, 5 possible 3rd place finishers, means 7 • 6 • 5 = 210 arrangements

Your turn: How many possible 1st, 2nd & 3rd place

arrangements are possible with 10 yachts?

!

( )!n r

nP

n r

7 3

7! 7!210

(7 3)! 4!P

720

6-7: PERMUTATIONS AND COMBINATIONS

When the order doesn’t matter, we use combinations

The only difference (mathematically) between nCr and nPr is the addition of r! in the denominator Example:

Evaluate

!

!( )!n r

nC

r n r

12 3

12! 12!

3!(12 3)! 3! 9!

12 11 10 9 8 7 6 5 4 3 2 1

3 2 1 9 8 7 6 5 4 3 2 1

12 11 10220

3 2 1

C

6-7: PERMUTATIONS AND COMBINATIONS

Example #4: A reading list for a course in world literature has 20 books on it. In how many ways can you choose four books to read?

Answer: 20 books, choosing 4 = 20C4 = 4845 Your turn

Evaluate 10C5

Of 20 books, in how many ways can you choose seven books?

252

77,520 ways

6-7: PERMUTATIONS AND COMBINATIONS

Example: Ten candidates are running for three seats in the student government. You may vote for as many as three candidates. In how many ways can you vote for three or fewer candidates?

Answer: If you vote for three people, it’s 10C3 = 120 ways

If you vote for two people, it’s 10C2 = 45 ways

If you vote for on person, it’s 10C1 = 10 ways

If you vote for no one, it’s 10C0 = 1 way 120 + 45 + 10 + 1 = 176 different ways

Your turn: In how many ways can you vote for five or fewer people?

638 ways

6-7: PERMUTATIONS AND COMBINATIONS

Worksheet Problems 1 – 27 Odd problems