Section 5.5 – Counting Techniques
Objectives
- Solve counting problems using the Multiplication rule
- Solve counting problems using permutations
- Solve counting problems using combinations
- Solve counting problems involving permutations with non-distinct items
- Compute probabilities involving permutations and combinations
Multiplication Rule of Counting
Example: A man has eight shirts and four ties. Assuming they all match, how many shirt and tie combinations can he wear?
Example: Texas license plates allow for three letters followed by four digits. How many license plates can be formed if letters and digits are allowed to repeat?
______
If letters and digits are not allowed to repeat
______
Factorials
On calculator: (MATH, PRB, #4)
Example:
5! = 10! =
==
N items can be arranged in N! different ways.
Examples:
a)You are car shopping and there are 8 different Nissan dealerships in the Dallas Metroplex. If you want to visit all 8 dealerships, how many different routes are possible?
b)How many different ways can 7 different books be arranged on a shelf?
c)How many ways can 15 students be lined up?
Permutations
With permutations, order is important!
On calculator: (MATH, PRB, #2)
Examples:
a)How many ways can a President, Vice-President, Secretary, and Treasurer be chosen from 10 committee members?
b)Lisa has 15 framed pictures. She wants to pick 5 of them to hang on the wall. How manydifferent ways can she hang 5 pictures on the wall?
c)How many ways can the horses in a 10-horse race finish first, second, and third?
Combinations
With combinations, order is NOT important!
On the calculator: (MATH, PRB, #3)
Examples:
a)How many ways can you pick 3 committee members from a group of 10 people?
b)How many random samples of size 4 can be obtained from a population whose size is 20?
Permutations with Non-Distinct Items
Example: How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red?
Computing Probabilities using Counting Techniques:
Remember
Examples:
a)Lotto Texas: Pick 6 numbers from 1 – 54 and the order does not matter.
How many number combinations are there?
What is the probability of winning Lotto Texas?
b)Outside a home, there is a keypad that can be used to open the garage if the correct four-digitcode is entered. The numbers on the keypad are 0 – 9.
How many codes are possible?
What is the probability that a burglar would enter the correct code on the first try?
c)Through a manufacturing error, three cans marked “Coca-Cola” were accidentally filled with “Diet Coke” and placed into a 12-pack. Suppose that three cans are randomly selected from the 12-pack. Calculate the following probabilities:
P (1 can contains Diet Coke)
P (2 cans contain Diet Coke)
P (all 3 cans contain Diet Coke)
d)Suppose you have received a shipment of 100 TVs. You don’t know it, but 6 of the TVs are defective. To determine if you are going to accept the shipment, you randomly select 5 TVs and test them. If all 5 TVs work, you will accept the shipment, otherwise, you will reject it. What is theprobability that you accept the shipment?
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