10.2 Combinations and Binomial Theorem

1
10.2 Combinations and Binomial Theorem
What you should learn
Goal
1
Use Combinations to count the number of ways an
event can happen.
Goal
2
Use the Binomial Theorem to expand a binomial
that is raised to a power.
10.2 Combinatins and Binomial Theorem
2
In the last section we learned counting problems
where order was important

• For other counting problems where order is NOT
  important like cards, (the order youre dealt is
  not important, after you get them, reordering
  them doesnt change your hand)
• These unordered groupings are called Combinations

12.2 Combinatins and Binomial Theorem
3
A Combination is a selection of r objects from a
group of n objects where order is not important
12.2 Combinatins and Binomial Theorem
4
Combination of n objects taken r at a time

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

12.2 Combinatins and Binomial Theorem
5

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

2
12.2 Combinatins and Binomial Theorem
6
Finding Combinations

• In a standard deck of 52 cards there are 4 suits
  with 13 of each suit.
• If the order isnt important how many different
  5-card hands are possible?
• The number of ways to draw 5 cards from 52 is

2,598,960
7
In how many of these hands are all 5 cards the
same suit?

• You need to choose 1 of the 4 suits and then 5 of
  the 13 cards in the suit.
• The number of possible hands are

12.2 Combinatins and Binomial Theorem
8
How many 7 card hands are possible?

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

12.2 Combinatins and Binomial Theorem
9

• When finding the number of ways both an event A
  and an event B can occur, you multiply.
• When finding the number of ways that an event A
  OR B can occur, you .

12.2 Combinatins and Binomial Theorem
10
Deciding to ADD or MULTIPLY

• A restaurant serves omelets. They offer 6
  vegetarian ingredients and 4 meat ingredients.
• You want exactly 2 veg. ingredients and 1 meat.
  How many kinds of omelets can you order?

12.2 Combinatins and Binomial Theorem
11
Suppose you can afford at most 3 ingredients

• How many different types can you order?
• You can order an omelet with 0, or 1, or 2, or 3
  items and there are 10 items to choose from.

12.2 Combinatins and Binomial Theorem
12

• Counting problems that involve at least or at
  most sometimes are easier to solve by
  subtracting possibilities you dont want from the
  total number of possibilities.

12.2 Combinatins and Binomial Theorem
13
Subtracting instead of adding

• A theatre is having 12 plays. You want to attend
  at least 3. How many combinations of plays can
  you attend?
• You want to attend 3 or 4 or 5 or or 12.
• From this section you would solve the problem
  using
• Or

14

• For each play you can attend you can go or not
  go.
• So, like section 10.1 it would be
  222222222222 212
• And you will not attend 0, or 1, or 2.
• So

15
(No Transcript)
16
The Binomial Theorem

• 0C0
• 1C0 1C1
• 2C0 2C1 2C2
• 3C0 3C1 3C2 3C3
• 4C0 4C1 4C2 4C3 4C4
• Etc

12.2 Combinatins and Binomial Theorem
17
Pascal's Triangle!

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• Etc
• This describes the coefficients in the expansion
  of the binomial (ab)n

18

• (ab)2 a2 2ab b2 (1 2 1)
• (ab)3 a3(b0)3a2b13a1b2b3(a0)
  (1 3 3 1)
• (ab)4 a44a3b6a2b24ab3b4 (1 4 6 4 1)
• In general

12.2 Combinatins and Binomial Theorem
19
(ab)n (n is a positive integer)

• nC0anb0 nC1an-1b1 nC2an-2b2 nCna0bn

12.2 Combinatins and Binomial Theorem
20
(a3)5

• 5C0a5305C1a4315C2a3325C3a2335C4a1345C5a035
• 1a5 15a4 90a3 270a2 405a 243

12.2 Combinatins and Binomial Theorem
21
Assignment
12.2 Combinatins and Binomial Theorem