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Wize University Statistics Textbook > Counting Techniques

Counting Techniques

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Fundamental Counting Principle

Suppose that an experiment has mm stages.
  • Let ni=n_i= # of outcomes in a given stage ii
  • Example: n3=6n_3=6 means there are 6 possible outcomes in the 3rd stage.
  • In the 1st stage there are n1n_1 outcomes, in the 2nd stage there are n2n_2 outcomes, in the third stage there are n3n_3 outcomes, ... in the mthm^{th} (final) stage there are nmn_m outcomes.
To calculate the total number of different possible outcomes when only one outcome occurs from each stage:

Outcomes=(n1)(n2)(n3)...(nm)\displaystyle\boxed{{Outcomes = (n_1)(n_2)(n_3)...(n_m)}}

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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.

n1=2n2=3\begin{aligned}n_1=2\\ n_2=3\end{aligned}

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


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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.

n1=3n2=2n3=1n4=4\begin{aligned}n_1=3\\ n_2=2\\ n_3=1\\ n_4=4 \end{aligned}
(3 drinks)(2 burgers)(1 fries)(4 dessserts)=24 options \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.

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Permutations and Combinations

Permutations

When solving for permutations, the order matters.

The number of ways we can arrange in order rr objects from a list of nn distinct objects is:

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

Wize Tip
Button often used on a calculator: nPr⁡\displaystyle\boxed{n\Pr}

Example #1
In how many ways can 10 employees fill 3 different shifts (i.e. morning, afternoon, evening)?

Order matters (permutation)
n=10n=10 employees
r=3r=3 shifts (in a specific order)

10P3=720 ways_{10}P_{3}=720\ ways


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Combinations

When solving for combinations, the order does not matter.

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

nCr=n!r!(n−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.
n!(n−r)!>n!r!(n−r)! \frac{n!}{\left(n-r\right)!}>\frac{n!}{r!\left(n-r\right)!}\
if
r>1r>1

Wize Concept
If r=1r=1 then nPr=nCr_nP_r=_nC_r

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Wize Tip
On a calculator this is the button nCr\displaystyle\boxed{nCr}

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=20n=20 participants to choose from
r=3r=3 participants to be chosen

10C3=(103)=120_{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?
There are 13 courses available for you to take next semester. How many ways can you enrol in 4 courses?

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More Combination Rules

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

AND=MultiplyOR=Add\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?

12C2+12C3=66+220=286_{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?

(12C2)(8C3)=(66)(56)=3696\left(_{12}C_{2}\right)\left(_{8}C_{3}\right)=\left(66\right)\left(56\right)=3696

(c) select four men from a group of men or one woman from a group of women?

12C4+8C1=495+8=503_{12}C_{4}+_{8}C_{1}=495+8=503

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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.

(2C1)(4C2)=(2)(6)=12\left(_{2}C_{1}\right)\left(_{4}C_{2}\right)=\left(2\right)\left(6\right)=12
Using a tree diagram could help!


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(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.

(2C1)(4P2)=(2)(12)=24\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.

Practice: Permutation or Combination?

For each question, determine if it is a permutation or a combination and select the correct numerical answer.
(a) You have homework to do for your history, chemistry, math, and geography classes. How many different ways can you do all four homework? (2 answers)
Extra Practice
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
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
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?