Wize University Statistics Textbook > Counting Techniques

Counting Techniques

Fundamental Counting Principle

Suppose that an experiment has $m$ stages.

Let $n_i=$ # of outcomes in a given stage $i$
Example: $n_3=6$ means there are 6 possible outcomes in the 3rd stage.
In the 1st stage there are $n_1$ outcomes, in the 2nd stage there are $n_2$ outcomes, in the third stage there are $n_3$ outcomes, ... in the $m^{th}$ (final) stage there are $n_m$ outcomes.

To calculate the total number of different possible outcomes when only one outcome occurs from each stage:

\displaystyle\boxed{{Outcomes = (n_1)(n_2)(n_3)...(n_m)}}

Example #1

Yyvonne is taking her dog to the dog spa. The spa package includes:

Choose one of: (A) bubble bath or (B) pure rain experience; and
Choose one of: (1) pawdicure, (2) massage, (3) Reiki

Determine how many possible treatment combos she can select from.

$\begin{aligned}n_1=2\\ n_2=3\end{aligned}$

\left(2\right)\left(3\right)=6\ combos

Example #2

For lunch, Jacob can pick:

Pepsi, Coca-Cola, or Sprite as a drink;
a chicken or beef burger;
he has to have fries; and
one of 4 possible desserts.

Determine how many possible meal combos he can select from.

$\begin{aligned}n_1=3\\ n_2=2\\ n_3=1\\ n_4=4 \end{aligned}$

$\left(3\ drinks\right)\left(2\ burgers\right)\left(1\ fries\right)\left(4\ dessserts\right)=24\ options\$

It is simple to count when you only choose one item from each stage.

Permutations and Combinations

Permutations

When solving for permutations, the order matters.

The number of ways we can arrange in order $r$ objects from a list of $n$ distinct objects is:

\displaystyle\boxed{_{n}P_{r}=\frac{n!}{\left(n-r\right)!}}

Example #1

In how many ways can 10 employees fill 3 different shifts (i.e. morning, afternoon, evening)?

Order matters (permutation)

$n=10$ employees

$r=3$ shifts (in a specific order)

$_{10}P_{3}=720\ ways$

Combinations

When solving for combinations, the order does not matter.

The total number of ways to choose $r$ objects from a set containing $n$ distinct objects is calculated using $\displaystyle \binom{n}{r}~ \text{ or }~_nC_r$ :

\displaystyle\boxed{_{n}C_{r}=\frac{n!}{r!\left(n-r\right)!}}

Unlike permutations, combinations do not care about the order of the items.
There are fewer combinations than permutations.

\frac{n!}{\left(n-r\right)!}>\frac{n!}{r!\left(n-r\right)!}\

r>1

Example #1

From a group of 10 participants, how many ways can 3 people be selected to participate in a survey?

Order does not matter (combination)

$n=20$ participants to choose from

$r=3$ participants to be chosen

$_{10}C_3=\binom{10}{3}=120$

Santa has 4 reindeers, Prancer, Dancer, Raver, and Bender, and he wants to have 3 fly his sleigh on Christmas Eve. He always has his reindeer fly in a single-file line: front, middle, back. How many different ways can he arrange his reindeer?

I don't know

There are 13 courses available for you to take next semester. How many ways can you enrol in 4 courses?

715

2,704

17,160

I don't know

More Combination Rules

You can apply the addition rule and multiplication rule to combinations:

\begin{array}{c} AND&=&Multiply\\ OR&=&Add \end{array}

Examples

We have a group of 12 men and 8 women. In how many ways can we...

(a) select two or three men from a group of men?

$_{12}C_2+_{12}C_3=66+220=286$

(b) select two men from a group of men and three women from a group of women?

$\left(_{12}C_{2}\right)\left(_{8}C_{3}\right)=\left(66\right)\left(56\right)=3696$

$_{12}C_{4}+_{8}C_{1}=495+8=503$

Example: More Combination Rules

(a) Yyvonne is taking her dog to the dog spa. The spa package includes:

Choose one of: (A) bubble bath or (B) pure rain experience; and
Choose two of: (1) pawdicure, (2) massage, (3) Reiki, (4) acupuncture (order is random)

Determine how many possible treatment combos she can select from.

\left(_{2}C_{1}\right)\left(_{4}C_{2}\right)=\left(2\right)\left(6\right)=12

Using a tree diagram could help!

(b) Suppose the spa package has been modified:

Choose one of: (A) bubble bath or (B) pure rain experience; and
Choose two of: (1) pawdicure, (2) massage, (3) Reiki, (4) acupuncture (must specify order)

Determine how many possible treatment combos she can select from.

\left(_{2}C_{1}\right)\left(_{4}P_{2}\right)=\left(2\right)\left(12\right)=24

There are 7 candidates from the West and 3 candidates from the East. Three of them will be elected.

A) In how many ways can we elect 3 candidates?B) In how many ways can we elect 2 candidates from the West and 1 candidate from the East?C) In how many ways can all 3 elected be from the West or from the East?

I don't know

Extra Practice

Counting Techniques

Twin Pines Country Club will issue membership ID's in this format: A12. It begins with a letter (A to Z) followed by two digits. Due to superstition, it will not use the digit "4" and the number "13". In addition, "69" will not be used because Twin Pines is a fancy club. How many unique membership ID's can be issued?

Counting Techniques

Santa has 4 reindeers, Prancer, Dancer, Raver, and Bender, and he wants to have 3 fly his sleigh on Christmas Eve. He always has his reindeer fly in a single-file line. How many different ways can he arrange his reindeer?

Counting Techniques

There are fifty candidates on “The Price is Right”. The announcer randomly calls 4 contestants to “come on down”. How many different orders of contestants could there be for the first announcement?

Counting Techniques

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Fundamental Counting Principle

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Permutations and Combinations

Permutations

Combinations

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More Combination Rules

Examples

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Example: More Combination Rules

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Practice: Permutation or Combination?

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Extra Practice