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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