12.2 Combinations 12.2 Combinations and Binomial and Binomial TheoremTheorem

p. 708p. 708What is a combination?What is a combination?

How is it different from a permutation?How is it different from a permutation?What is the formula for What is the formula for nnCCrr??What is Pascal’s Triangle?What is Pascal’s Triangle?

What is the Binomial Theorem?What is the Binomial Theorem?How is Pascal’s Triangle connected to the How is Pascal’s Triangle connected to the

binomial theorem?binomial theorem?

In the last section we In the last section we learned counting problems learned counting problems where order was where order was importantimportant

For other counting problems where For other counting problems where order is NOT important like cards, order is NOT important like cards, (the order you’re dealt is not (the order you’re dealt is not important, after you get them, important, after you get them, reordering them doesn’t change reordering them doesn’t change your hand)your hand)

These unordered groupings are These unordered groupings are called called CombinationsCombinations

A A CombinationCombination is a is a selection of “selection of “r”r” objects objects from a group of “from a group of “n”n” objects objects where order is not where order is not importantimportant

Combination of n objects Combination of n objects taken r at a timetaken r at a time

The number of combinations of r The number of combinations of r objects taken from a group of n objects taken from a group of n distinct objects is denoted by distinct objects is denoted by nnCCrr and is:and is:

!)!(

!

rrn

nCrn

For instance, the number of For instance, the number of combinations of 2 objects taken combinations of 2 objects taken from a group of 5 objects isfrom a group of 5 objects is

101*2*1*2*3

1*2*3*4*5

!2)!25(

!525

C

2

Finding CombinationsFinding Combinations

In a standard deck of 52 cards In a standard deck of 52 cards there are 4 suits with 13 of each there are 4 suits with 13 of each suit.suit.

If the order isn’t important how If the order isn’t important how many different 5-card hands are many different 5-card hands are possible?possible?

The number of ways to draw 5 The number of ways to draw 5 cards from 52 is cards from 52 is !5!*47

!47*48*49*50*51*52

!5)!552(

!52552

C

= 2,598,960

In how many of these In how many of these hands are all 5 cards the hands are all 5 cards the same suit?same suit?

You need to choose 1 of the 4 suits You need to choose 1 of the 4 suits and then 5 of the 13 cards in the and then 5 of the 13 cards in the suit.suit.

The number of possible hands are:The number of possible hands are:5148

!5!*8

!8*9*10*11*12*13*

!1!*3

!3*4

!5!*8

!13*

!1!*3

!4* 51314 CC

How many 7 card hands How many 7 card hands are possible?are possible?

How many of these hands have all How many of these hands have all 7 cards the same suit?7 cards the same suit?

560,784,133!7!*45

!52752 C

6864* 71314 CC

When finding the number of When finding the number of ways both an event A ways both an event A andand an event B can occur, you an event B can occur, you multiply.multiply.

When finding the number of When finding the number of ways that an event A ways that an event A OROR B B can occur, you +.can occur, you +.

Deciding to + or *Deciding to + or *

A restaurant serves omelets. They A restaurant serves omelets. They offer 6 vegetarian ingredients and offer 6 vegetarian ingredients and 4 meat ingredients.4 meat ingredients.

You want exactly 2 veg. You want exactly 2 veg. ingredients and 1 meat. How ingredients and 1 meat. How many kinds of omelets can you many kinds of omelets can you order?order?

604*15!1!3

!4*

!2!4

!6* 1426 CC

Suppose you can afford at Suppose you can afford at most 3 ingredientsmost 3 ingredients

How many different types can you How many different types can you order?order?

You can order an omelet with 0, or You can order an omelet with 0, or 1, or 2, or 3 items and there are 10 1, or 2, or 3 items and there are 10 items to choose from.items to choose from.

17612045101310210110010 CCCC

Counting problems that Counting problems that involve ‘at least’ or ‘at involve ‘at least’ or ‘at most’ sometimes are most’ sometimes are easier to solve by easier to solve by subtracting possibilities subtracting possibilities you don’t want from the you don’t want from the total number of total number of possibilities.possibilities.

Subtracting instead of Subtracting instead of adding:adding:

A theatre is having 12 plays. You A theatre is having 12 plays. You want to attend at least 3. How many want to attend at least 3. How many combinations of plays can you combinations of plays can you attend?attend?

You want to attend 3 or 4 or 5 or … or You want to attend 3 or 4 or 5 or … or 12.12.

From this section you would solve the From this section you would solve the problem using:problem using:

Or……Or……1212512412312 ... CCCC

For each play you can attend For each play you can attend you can go or not go. you can go or not go.

So, like section 12.1 it would So, like section 12.1 it would be 2*2*2*2*2*2*2*2*2*2*2*2 be 2*2*2*2*2*2*2*2*2*2*2*2 =2=21212

And you will not attend 0, or And you will not attend 0, or 1, or 2.1, or 2.

So:So:

4017)66121(4096)(2 21211201212 CCC

The Binomial TheoremThe Binomial Theorem

00CC00

11CC0 10 1CC11

22CC00 22CC11 22CC22

33CC00 33CC11 33CC2 32 3CC3 3

44CC00 44CC11 44CC22 44CC33 44CC44

Etc…Etc…

Pascal's Triangle!Pascal's Triangle!

11 1 11 1

1 2 11 2 1 1 3 3 11 3 3 1

1 4 6 4 11 4 6 4 1 1 5 10 10 5 11 5 10 10 5 1

Etc…Etc… This describes the coefficients in the This describes the coefficients in the

expansion of the binomial (a+b)expansion of the binomial (a+b)nn

(a+b)(a+b)22 = a = a22 + 2ab + b + 2ab + b22 (1 2 1) (1 2 1)(a+b)(a+b)33 = =

aa33(b(b00)+3a)+3a22bb11+3a+3a11bb22+b+b33(a(a00) ) (1 3 3 1) (1 3 3 1)

(a+b)(a+b)44 = = aa44+4a+4a33b+6ab+6a22bb22+4ab+4ab33+b+b4 4 (1 4 6 (1 4 6 4 1)4 1)

In general…In general…

(a+b)(a+b)nn (n is a positive (n is a positive integer)=integer)=

nnCC00aannbb0 0 + + nnCC11aan-1n-1bb1 1 + + nnCC22aan-2n-2bb22 + …+ + …+

nnCCnnaa00bbnn

==

n

r

rrnrn baC

0

(a+3)(a+3)55 = =

55CC00aa553300++55CC11aa443311++55CC22aa333322++55CC33aa223333++55CC44aa113344++55CC55aa003355==

1a1a5 5 + 15a+ 15a4 4 + 90a+ 90a3 3 + 270a+ 270a2 2 + + 405a + 243405a + 243

(a+3)(a+3)55 = =1a1a553300+5a+5a443311+10a+10a333322+10a+10a223333+5+5

aa113344+1a+1a003355

1a1a5 5 + 15a+ 15a4 4 + 90a+ 90a3 3 + 270a+ 270a2 2 + + 405a + 243405a + 243

111 11 1

1 2 11 2 11 3 3 11 3 3 1

1 4 6 4 11 4 6 4 11 5 10 10 5 11 5 10 10 5 1

What is a combination?What is a combination?

A selection of objects where order is not important.A selection of objects where order is not important. How is it different from a permutation?How is it different from a permutation?

In other counting problems, order is not important.In other counting problems, order is not important. What is the formula for What is the formula for nnCCrr?? What is Pascal’s Triangle?What is Pascal’s Triangle?

An arrangement of the values of An arrangement of the values of nnCCr r in a triangular in a triangular pattern.pattern.

What is the Binomial Theorem?What is the Binomial Theorem?

A formula that gives the coefficients in the raising of A formula that gives the coefficients in the raising of (a+b) to a power.(a+b) to a power.

How is Pascal’s Triangle connected to the binomial How is Pascal’s Triangle connected to the binomial theorem?theorem?

Pascal’s Triangle gives the same coefficients as the Pascal’s Triangle gives the same coefficients as the Binomial Theorem.Binomial Theorem.

!!

!

rrn

n

Assignment Assignment 12.212.2

P. 712P. 712

19-39 odd, 45-57 19-39 odd, 45-57 odd, skip 51odd, skip 51