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Chapter 11. Section 11.8. Exercise #1. A parent teacher committee consisting of four people is to be formed from 20 parents and five teachers. Find the probability that the committee will consist of the following. (Assume that the selection will be random.). (a) All teachers

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A parent teacher committee consisting of four people is to be formed from 20 parents and five teachers. Find the probability that the committee will consist of the following. (Assume that the selection will be random.)

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25!

25C4

=

4!(25 – 4)!

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25!

25C4

=

4!(21!)

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25 24 23 22 21!

25C4

=

4 3 2 1 21!

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25C4

= 12,650

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5!

5C4

=

4!(5 – 4)!

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5!

5C4

=

4!(1!)

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5 4!

5C4

=

4! 1

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5C4

= 5

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

n(all teachers)

P(all teachers) =

n(all selections)

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

5

P(all teachers) =

12,650

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

1

P(all teachers) =

2,530

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

P(all teachers) = 0.0004

5!

20!

5C2

20C2

=

2!(3!)

2!(18!)

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

5 4

20 19

5C2

20C2

=

2 1

2 1

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

20

380

5C2

20C2

=

2

2

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

5C2

20C2

= 10 190

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

5C2

20C2

= 1,900

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents): 1,900

P(2 teachers and 2 parents) =

n(2 teachers and 2 parents)

n(all selections)

n(all selections): 12,650

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents): 1,900

n(all selections): 12,650

P(2 teachers and 2 parents) =

1,900

12,650

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

P(2 teachers and 2 parents) =0.15

20!

20C4

=

4!(16!)

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents):

20 19 18 17

20C4

=

4 3 2 1

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents):

116,280

20C4

=

24

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents): 4,845

n(all parents)

P(all parents):

n(all selections)

n(all selections): 12,650

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents): 4,845

n(all selections): 12,650

4,845

P(all parents):

12,650

n(1 teacher and 3 parents):

5!

20!

5C1

20C3

=

1!(4!)

3!(17!)

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

5

20 19 18

5C1

20C3

=

1

3 2 1

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

5

6,840

5C1

20C3

=

1

6

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

34,200

5C1

20C3

=

6

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

5C1

20C3

= 5,700

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents): 5,700

P(1 teacher and 3 parents) =

n(1 teacher and 3 parents)

n(all selections)

n(all selections): 12,650

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents): 5,700

n(all selections): 12,650

P(1 teacher and 3 parents) =

5,700

12,650

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

P(1 teacher and 3 parents) =0.450

A city council consists of 10 members. Four are Republicans, three are Democrats, and three are Independents. If a committee of three is to be selected, find the probability of selecting

All Republicans

All Democrats

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10!

10C3

=

3!(10 – 3)!

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10!

10C3

=

3!(7!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10 9 8 7!

10C3

=

3 2 1 7!

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

720

10C3

=

6

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10C3

= 120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4!

4C3

=

3!(4 – 3)!

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4!

4C3

=

3!(1!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4 3!

4C3

=

3! 1

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4C3

= 4

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R): 4

n(all R)

P(all R) =

n(all selections)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R): 4

4

P(all R) =

120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R): 4

1

P(all R) =

30

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

P(all R) = 0.033

3!

3C3

=

3!(3 – 3)!

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D):

3!

3C3

=

3!(0)!

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D):

3C3

= 1

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D): 1

n(all D)

P(all D) =

n(all selections)

n(all selections): 120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D): 1

n(all selections): 120

1

P(all D) =

120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

P(all D) = 0.0083

4!

3!

3!

4C1

3C1

3C1

=

1!(3!)

1!(2!)

1!(2!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I):

4C1

3C1

3C1

= 4 3 3

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I):

4C1

3C1

3C1

= 36

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I): 36

P(one R, one D, one I) =

n(one R, one D, one I)

n(all selections)

n(all selections): 120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I): 36

n(all selections): 120

P(one R, one D, one I) =

36

120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

P(one R, one D, one I) =0.300

n(two D and one I):

3!

3!

3C2

3C1

=

2!(1!)

1!(2!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I):

3C2

3C1

= 3 3

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I):

3C2

3C1

= 9

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I): 9

n(two D and one I)

P(two D and one I) =

n(all selections)

n(all selections): 120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I): 9

n(all selections): 120

9

P(two D and one I) =

120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

P(two D and one I) = 0.075

n(one I and two R):

3!

4!

3C1

4C2

=

1!(2!)

2!(2!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

4 3

3C1

4C2

= 3

2 2

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

12

3C1

4C2

= 3

2

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

3C1

4C2

= 3 6

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

3C1

4C2

= 18

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R): 18

n(one I and two R)

P(one I and two R) =

n(all selections)

n(all selections): 120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R): 18

n(all selections): 120

18

P(one I and two R) =

120

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

P(one I and two R) =0.15

Find the probability of selecting three science books and four math books from eight science books and nine math books. The books are selected at random.

8!

9!

8C3

9C4

=

3!(5!)

4!(5!)

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

8 7 6 5!

9 8 7 6 5!

8C3

9C4

=

3 2 1 5!

4 3 2 1 5!

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

336

3024

8C3

9C4

=

6

24

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

8C3

9C4

= 56 126

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

8C3

9C4

= 7056

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

17!

17C7

=

7!(10!)

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

17 16 15 14 13 12 11 10!

17C7

=

7 6 5 4 3 2 1 10!

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

98,017,920

17C7

=

5040

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

17C7

= 19,448

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(3 S and 4 M)

P(3 S and 4 M) =

n(all selections)

n(all selections): 19,448

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections): 19,448

7,056

P(3 S and 4 M) =

19,448

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

P(3 S and 4 M) =0.363

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