Chapter 12 – Probability and Statistics12.2 – Permutations and Combinations

12.2 – Permutations and Combinations• Today we will learn how to:• Solve problems involving permutations

• Solve problems involving combinations

12.2 – Permutations and Combinations• Permutation – when a group of objects or people are

arranged in a certain (particular) order• The order of the objects is very important

• Linear permutation – the arrangement of objects or people in a line

• The number of permutations of n distinct objects taken r at a time is given by• P (n, r) = n! (n – r)!• nPr

12.2 – Permutations and Combinations• Example 1• Eight people enter the Best Pie contest. How many ways can

blue, red, and yellow ribbons be awarded?

12.2 – Permutations and Combinations• Suppose you want to rearrange the letters of the word

geometry to see if you can make a different word. If the two es were not identical, the eight letters in the word could be arranged in P(8, 8) ways. To account for the identical es, divide P(8, 8) by the number of arrangements of e. The two es can be arranged in P(2, 2) ways.• P(8, 8) = 8! P(2, 2) 2!

= 8 · 7 · 6 · 5 · 4 · 3 · 2! or 20,160 2!

12.2 – Permutations and Combinations• Permutations with Repetitions• The number of permutations of n objects of which p are alike and

q are alike is • n! p! q!

12.2 – Permutations and Combinations• Example 2• How many different ways can the letters of the word BANANA be

arranged?

12.2 – Permutations and Combinations• Combinations – an arrangement or selection of objects in

which the order is not important• The number of combinations of n objects taken r at a time is

written C(n, r) or nCr

• You know that there are P(n, r) ways to select r objects from a group of n if the order is important. There are r! ways to order the r objects that are selected, so there are r! permutations that are all the same combination. Therefore:• C(n, r) = P(n, r) or n! . r! (n – r)! r!

12.2 – Permutations and Combinations• Combinations• The number of combinations of n distinct objects taken r at a

time is given by• C (n, r) = n! . (n – r)! r!

12.2 – Permutations and Combinations• Example 3• Five cousins at a family reunion decide that three of them will go

to pick up a pizza. How many ways can they choose the three who will go?

12.2 – Permutations and Combinations• In more complicated situations, you may need to multiply

combinations and/or permutations

12.2 – Permutations and Combinations• Example 4• Six cards are drawn from a standard deck of cards. How many

hands consist of two hearts and four spades?

12.2 – Permutations and Combinations

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