Combinations - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Counting Techniques:

Combinations

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Combinations

Typical situation: How many 5-card hands

can be made from a deck of 52 cards?

Definition: r-combination of a set of n elements

is a subset of r of the n elements.

• The number of all r-combinations

of a set of n elements denoted

and read “ n choose r ”.

In the example above, want to find C 52,.

C n r Cn r r

n , ( , ), , 

  



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Formula for computing C(n,r)

Suppose we want to compute P(n,r).

Constructing an r-permutation from a set of n elements

can be thought as a 2-step process:

Step 1: Choose a subset of r elements;

Step 2: Choose an ordering of the r-element subset.

Step 1 can be done in C(n,r) different ways.

Step 2 can be done in r! different ways.

( regardless of how the step 1 was performed)

Based on the multiplication rule, P(n,r) = C(n, r) ∙ r!

Thus,

r n r

n

r

P n r

C n r

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Examples on Combinations

 The number of different 5-card hands

from a deck of 52 cards:

 4 members from a group of 11

are supposed to work as a team on a project.

Q: How many distinct 4-person teams can be chosen?

A:

C =

C =

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Examples on Combinations

 Suppose that 3 cars in a production run of 40 are defective.

A sample of 4 is to be selected to be checked for defects.

Questions:

1. How many different samples can be chosen?

2. How many samples will contain

exactly one defective car?

3. What is the probability that a randomly chosen sample will contain exactly one defective car?

4. How many samples will contain

at least one defective car?

Solution:

1. C(40, 4) = (40∙39∙38∙37) / (1∙2∙3∙4) = 91,

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Examples on Combinations

2. How many samples will contain

exactly one defective car?

Think of selecting a sample as a 2-step process:

Step 1: Choose the defective cars;

Step 2: Choose the good cars.

There are C(3,1) ways to choose 1 defective car.

There are C(37,3) ways to choose 3 good cars.

By the multiplication rule, the number of samples containing exactly 1 defective car is

C(3,1) ∙ C(37,3) = 3∙(37∙36∙35) / (1∙2∙3) = 23,

3. What is the probability that a randomly chosen sample will contain exactly one defective car?

The probability = 23,310 / 91,390 =.

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Some Advice about Counting

 Apply the multiplication rule if

▪ The elements to be counted

can be obtained through a multistep process;

▪ Each step is performed

in a fixed number of ways regardless of

how preceding steps were performed.

 Apply the addition rule if

▪ The set of elements to be counted

can be broken up into disjoint subsets.

 Note: Often a counting problem is solved by applying

both the multiplication and addition rules (and their variations) at different stages of the solution.

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Some Advice about Counting

 In any counting problem, make sure that

1. every element is counted;

2. no element is counted more than once.

( avoid double counting )

• When using the multiplication rule,

these directives become:

1. every outcome should appear as some branch of tree;

2. no outcome should appear

on more than one branch of tree.

• When using the addition rule, the directives become:

1. every outcome should be in some subset;

2. the subsets should be disjoint. Docsity.com