Warm UpWarm Up

Suppose we wanted to sit six people, three men Suppose we wanted to sit six people, three men and three women in a row of six chairs. How and three women in a row of six chairs. How many ways can we seat them if we must start many ways can we seat them if we must start with a woman and then alternate women and with a woman and then alternate women and men?men?

7272 How many ways can we form a license plate if How many ways can we form a license plate if

we want three digits followed by two letters?we want three digits followed by two letters?676000676000

12.3 Permutations and 12.3 Permutations and CombinationsCombinations

Calculate the number of permutations of n Calculate the number of permutations of n objects taken r at a time.objects taken r at a time.

Use factorial notation to represent the Use factorial notation to represent the number of permutations of a set of objects.number of permutations of a set of objects.

Calculate the number of combinations of n Calculate the number of combinations of n objects taken r at a time.objects taken r at a time.

Apply the theory of permutations and Apply the theory of permutations and combinations to solve counting problems.combinations to solve counting problems.

Permutations are arrangements Permutations are arrangements of objects in a straight line.of objects in a straight line.

PermutationPermutation – an ordering of distinct – an ordering of distinct objects in a straight line. If we select r objects in a straight line. If we select r different objects from a set of n objects different objects from a set of n objects and arrange them in a straight line, this is and arrange them in a straight line, this is called a permutation of n objects taken r at called a permutation of n objects taken r at a time. The number of permutations of n a time. The number of permutations of n objects taken r at a time is denoted by objects taken r at a time is denoted by P(n,r).P(n,r).

How many permutations are there of the How many permutations are there of the letters a, b, c, and d?letters a, b, c, and d? 4 x 3 x 2 x 1 = 244 x 3 x 2 x 1 = 24 P(4, 4) = 24P(4, 4) = 24

How many permutations are there of the How many permutations are there of the letters a, b, c, d, e, f, and g if we take letters a, b, c, d, e, f, and g if we take the letters three at a time?the letters three at a time? 7 x 6 x 5 = 2107 x 6 x 5 = 210 P(7, 3) = 210P(7, 3) = 210

We use the factorial notation to We use the factorial notation to express P(n, r).express P(n, r).

If n is a counting number, the symbol n!, If n is a counting number, the symbol n!, called n factorial, stands for the product called n factorial, stands for the product n•(n – 1) • (n – 2) • (n – 3) • … • 2 • 1. We n•(n – 1) • (n – 2) • (n – 3) • … • 2 • 1. We define 0! = 1.define 0! = 1.

P(n, r) = P(n, r) = n! n!(n – r)!(n – r)!

In forming combinations, order In forming combinations, order is not importantis not important

If we choose r objects from a set of n If we choose r objects from a set of n objects, we say that we are forming a objects, we say that we are forming a combinationcombination of n objects taken r at a of n objects taken r at a time. We are only concerned with time. We are only concerned with choosing a set, but the order is not choosing a set, but the order is not important. The notation C(n, r) denotes important. The notation C(n, r) denotes the number of such combinations.the number of such combinations.

C(n, r) = C(n, r) = P(n, r)P(n, r) = = n! n! r!r! r! • (n – r)! r! • (n – r)!

Some good adviceSome good advice

In working with permutations and combinations, In working with permutations and combinations, we are choosing r different objects from a set of we are choosing r different objects from a set of n objects. The big difference is whether the n objects. The big difference is whether the order of the objects is important. If it is, then we order of the objects is important. If it is, then we are dealing with a permutation. If not, then we are dealing with a permutation. If not, then we are working with a combination.are working with a combination.

If a problem involves something other than If a problem involves something other than simply choosing different objects, then perhaps simply choosing different objects, then perhaps we should use the fundamental counting we should use the fundamental counting principle.principle.

You and fifteen of your friends have formed a You and fifteen of your friends have formed a company. A committee consisting of a company. A committee consisting of a president, vice president, and a three-president, vice president, and a three-member executive board need to be formed. member executive board need to be formed. How many different ways can this committee How many different ways can this committee be formed?be formed? Choose the president and vice president.Choose the president and vice president.

• The order is important. P(16, 2)The order is important. P(16, 2) Select the remaining 3 executive members.Select the remaining 3 executive members.

• The order is not important. C(14, 3)The order is not important. C(14, 3) We can do stage one followed by stage two by We can do stage one followed by stage two by

using the fundamental counting principle: using the fundamental counting principle:

P(16, 2) x C(14, 3) = 87,360 ways.P(16, 2) x C(14, 3) = 87,360 ways.

Warm UpWarm Up

A class has 10 boys and 12 girls. The teacher A class has 10 boys and 12 girls. The teacher wants to choose 4 boys and 3 girls to play 7up. wants to choose 4 boys and 3 girls to play 7up. How many ways is this possible?How many ways is this possible?

210 x 220 = 46,200210 x 220 = 46,200 A committee has 10 people. A president and A committee has 10 people. A president and

vice president needs to be chosen. Then a vice president needs to be chosen. Then a three person board must be picked. How many three person board must be picked. How many ways can this be done?ways can this be done?

90 x 56 = 5,04090 x 56 = 5,040

Pascal’s TrianglePascal’s Triangle

The rows of Pascal’s triangle The rows of Pascal’s triangle count the subsets of a set.count the subsets of a set.

Each entry in this triangle after row 1 is the sum Each entry in this triangle after row 1 is the sum of the two numbers immediately above it.of the two numbers immediately above it.

The nth row of Pascal’s triangle counts the The nth row of Pascal’s triangle counts the subsets of various sizes of an n-element set.subsets of various sizes of an n-element set. List the subsets of {1,2,3,4}: O, {1}, {2}, {3}, {4}, {1,2}, List the subsets of {1,2,3,4}: O, {1}, {2}, {3}, {4}, {1,2},

{1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}{2,3,4}, {1,2,3,4}

Note the sizes of each are 1 of size zero, 4 of size Note the sizes of each are 1 of size zero, 4 of size one, 6 of size two, 4 of size three, and 1 of size four. one, 6 of size two, 4 of size three, and 1 of size four. This pattern occurs as the fourth row in Pascal’s This pattern occurs as the fourth row in Pascal’s triangle.triangle.

The entries of Pascal’s triangle The entries of Pascal’s triangle are numbers of the form C(n, r).are numbers of the form C(n, r).

The rth entry of the nth row of Pascal’s The rth entry of the nth row of Pascal’s triangle is C(n, r).triangle is C(n, r). The fourth row of Pascal’s triangle is The fourth row of Pascal’s triangle is

1 4 6 4 11 4 6 4 1 Because the leftmost 1 is the zeroth entry in Because the leftmost 1 is the zeroth entry in

the fourth row, we can write it as C(4, 0). That the fourth row, we can write it as C(4, 0). That is to say, C(4, 0) = 1. The first entry in the is to say, C(4, 0) = 1. The first entry in the fourth row is 4, which means that C(4, 1) = 4.fourth row is 4, which means that C(4, 1) = 4.

Classwork/HomeworkClasswork/Homework

Classwork: Page 708 (7 – 15 odd, 19, 21, Classwork: Page 708 (7 – 15 odd, 19, 21, 37, 41)37, 41)

Homework: Page 708 (8 – 16 even, 20, Homework: Page 708 (8 – 16 even, 20, 22, 38, 42)22, 38, 42)

Classwork/HomeworkClasswork/Homework

Classwork: Page 708 (17, 23, 25, 27, 33, Classwork: Page 708 (17, 23, 25, 27, 33, 35, 39, 43, 45, 47, 49, 57, 59, 63)35, 39, 43, 45, 47, 49, 57, 59, 63)

Homework: Page 708 (18, 24, 26, 28, 34, Homework: Page 708 (18, 24, 26, 28, 34, 36, 40, 46, 50, 58)36, 40, 46, 50, 58)