Notes Over 12.2Finding CombinationsFinding Combinations

1. An ice cream shop has a choice of 10 toppings. In how many ways can you choose 3 different toppings for your ice cream?

rnC310

!10 !310 !3

!78910

!7201

123

3 4

Notes Over 12.2Finding CombinationsFinding Combinations

2. Out of a group of 30 people, 20 of which are women, how many different committees of 6 people can be chosen?

rnC630

!30 !630 !6

!24252627282930

!2493,7755

123456

7 9 13

Notes Over 12.2Finding CombinationsFinding Combinations

3. In exercise 2, determine the possible number of committees of 6 in which 4 are women and 2 are men.

rnC420

!20 ! 420 !4

!1617181920

!16

18,0252 1234

35 5

rnC210

!10 ! 210 !2

!8910

!8 12

Notes Over 12.2Using the Binomial TheoremUsing the Binomial Theorem

nba 0

0C ba nn

11

1C ba nn

22

2C bann

n

nnba0C rrn

n

rrn

baC

0

Notes Over 12.2Using the Binomial TheoremUsing the Binomial Theorem

41 .4 x04

041C x 13

141C x 22

241C x

31

341C x

4 x

Use the binomial theorem to write the binomial expansion.

40

441C x

3 4 x

!4!4 !0

!4!3 !1

2 6 x x 4 1

!4!2 !2

!4!1 !3

!4!0 !4

Notes Over 12.2Using the Binomial TheoremUsing the Binomial Theorem

52 .5 x

05

052C x 14

152C x 23

252C x

32

352C x

5 x

Use the binomial theorem to write the binomial expansion.

41

452C x

4 10 x

!5!5 !0

!5!4 !1

3 40 x 2 80 xx80

!5!3 !2

!5!2 !3

!5!1 !4

50

552C x

!5!0 !5

32

Notes Over 12.2Using the Binomial TheoremUsing the Binomial Theorem

4 .6 yx

04

04C yx 13

14C yx 22

24C yx

31

34C yx

4 x

Use the binomial theorem to write the binomial expansion.

40

44C yx

yx3 4

!4!4 !0

!4!3 !1

22 6 yx 3 4 xy 4y

!4!2 !2

!4!1 !3

!4!0 !4

Notes Over 12.2Using the Binomial TheoremUsing the Binomial Theorem

5 .7 yx

05

05C yx 14

15C yx 23

25C yx

32

35C yx

5 x

Use the binomial theorem to write the binomial expansion.

41

45C yxyx 45

!5!5 !0

!5!4 !1

23 01 yx 32 10 yx45xy

!5!3 !2

!5!2 !3

!5!1 !4

50

55C yx

!5!0 !5

5y

Notes Over 12.2Using the Binomial TheoremUsing the Binomial Theorem

7

5

4 ofexpansion the

in oft coefficien theFind .8

x

x

25

274C x

12

Use the binomial theorem to write the binomial expansion.

16

!7!5 !2

5 x5 336 x 336

Notes Over 12.2