COLLEGE PREP10.2 Use Combinations and the Binomial Theorem

Warm Up• 1) Evaluate 5!

• 2) Evaluate (4 – 2)! 3!

• 3) Evaluate 7P5

• 4) In how many ways can 6 people line up to buy tickets for a movie?

• 5) Find the number of distinguishable permutations of the letters in the word TENNESSEE.

• A combination is a selection or r objects from a group of n objects where the order is not important. It is denoted as

nCr = �!

��� !·�!

Parents have 10 books that they can read to their children this week. Five of the books are nonfiction and 5 of the books are fiction.

• If the order in which they read the books does not matter, how many different sets of four books can they choose?

10C4 = ��!

�!!= 210

• In how many groups of 4 books are all the books either nonfiction or fiction?

5C4+5C4 = �!

�!!+

�!

�!!=10

The student senate consists of 6 seniors, 5 juniors, 4 sophomores and 3 freshmen.

When finding the number of ways an event A and an event B can occur you multiply the combinations.

• How many different committees of exactly 2 seniors and 2 juniors be chosen?

When finding the number of ways an event A or and event B can occur you add the combinations.

• How many different committees of at most 4 students can be chosen?

• This means 1 or 2 or 3 or 4

When you see phrases such as at least or at most it is sometimes easier to subtract combinations instead of adding them. You subtract from the total possible.

You are going to toss 10 different coins. How many different ways will at least 4 of the coins show heads?

• You could add combinations for 4, 5, 6, 7, 8, 9, 10 or you could add up combinations for 1, 2, 3 and subtract it from the total.

There are two possible outcomes for each coin so the total possibilities would be 210.

• 210 – (10C1+10C2+10C3) = 849

Examples

Find the combinations:

• 10C6

• 7C2

• Shakespeare wrote 38 plays that can be divided into three genres. Of the 38 plays, 18 are comedies, 10 are histories, and 10 are tragedies.

• How many sets of exactly 3 tragedies and 2 histories can you read?

Pascal’s Triangle – an arrangement of values of

nCr in a triangular pattern in which each row corresponds with a value of n.

Binomial Theorem1=

1+= x

122++= xx

133 23+++= xxx

1464 234++++= xxxx

Write the coefficients of each expansion

1=

1+= x

122++= xx

133 23+++= xxx

1464 234++++= xxxx

What would 5)1( +x

11 11 2 11 3 3 11 4 6 4 1

Pascal’s Triangle (named after Blaise Pascal) represents the coefficients for a binomial expansion.

What do you notice about the diagonals?

From a collections of 7 baseball caps, you want to trade 3. Use Pascal’s triangle

to find the number of combinations of 3 caps that can be traded.

Binomial Theorem – there is a relationship between powers of binomials and

combinations. For example, if we want to expand the binomial (a + b)4

(a + b)4 = 4C0a4 + 4C1a

3b + 4C2a2 b2 + 4C3ab3 + 4C4b

4

(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3+ 1b4

Notice the coefficients are the same

numbers as the fourth row in Pascal’s

Triangle.

Notice the exponents on the first term

decrease by one and the exponents on the

second term increase by one.

Expand (x + y)6 using the binomial theorem.

Expand (3x – 2)4 using the binomial theorem.

Find the coefficient of x3y4 in (2x – y)7.

Use the formula:

nCr an - r br

n = 7, a = 2x, b = -y

• 7Cr(2x)7-r(-y)r.

• The term containing x3y4

is when r = 4 so we would have

• 7C4(2x)3(-y)4 = (35)(8x3)(y4) = 280x3y4

Homework

•Pg 694: 3-12 every other pair, 13-16, 19-32 every other pair, 33-34, 38-39, 49-50