Stats Boot Camp

Permutations & Combinations:

A Part of the Boot Camp for the Intro to Statistics Course

Each slide has its own narration. Click on the audio icon to start it.

Professor Friedman's Statistics Course by H & L Friedman is licensed under a �Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

This module is part of our Introduction to Statistics course. See more at https://stats.proffriedman.net/

Overview of this Module

The material in this part of the Stats Bootcamp is also contained in the Binomial Distribution Lecture

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• Permutations (arrangements)
• Combinations (collections)

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Permutations

Permutations. A permutation is a particular arrangement.

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• Example: How many ways can you arrange the letters A, B, and C?

Let’s try it. List all the possible arrangements (permutations):

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Answer:  6

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Permutations

• How did we get this? Let’s use 3 imaginary slots.

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• In slot one, we have three letters to choose from, A, B, or C. In slot two, we only have two letters left. Finally, in slot three, we only have one unused letter.

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Permutations

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• On your scientific calculator, you will use the nPr key.
• The P stands for permutations,
• n is the number of distinct objects you wish to arrange, and
• r is the number of slots or spaces.

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• Thus, in the above example, we have 3 objects to arrange in three slots, or, 3P3 = 6.

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Permutations

• The general formula for a permutation is:

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• [Sidebar] What is a Factorial?

n! is read as “n factorial”

where n! = n⋅(n-1)⋅(n-2)⋅ … ⋅(2)⋅(1) = n⋅(n-1)!

So, for example,

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

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When n = r, then nPn is simply = n!

[Note that 0! ≡ 1]

Permutations – Some Examples

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• How many ways can you assign five workers to five different tasks?

Answer: 5P5= 5! = 120

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• How many ways can you arrange 10 different books in your bookcase which has room for exactly 5 books?

Answer: 10P5= 10 x 9 x 8 x 7 x 6 = 30,240

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Permutations – More Examples

• How many ways can 8 cars line up single file in front of a toll booth?

Answer: 8P8 = 8! = 40,320

• How many ways can you arrange 12 guests around a rectangular table that has 12 chairs?

Answer: 12P12 = 12! = 479,000,600

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Follow up question: How many different potential family feuds can break out?

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Permutations & Combinations

• With permutations, the arrangement of the items is important.
• Each unique sequence is another permutation.
• Thus, ABC is different from BCA and both are different from CBA.

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• With combinations, however, ABC, BCA, and CBA are all the same. They are all part of the same combination.
• Example: How many different groups of 3 can be selected from 7 people? Say these people are named A, B, C, D, E, F, G. Note that once you select, say, B, D, and E, the six different arrangements you can make from them are irrelevant.

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Combinations

The formula for combinations is:

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Note that:

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You can use the nCr key on your calculator to solve any combination problem.

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Example: How many different groups of 3 can be selected from 7 people?

Answer: 7C3 = 7! / 3! 4! = 35

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Combinations

• Example: How many different hands can one draw from a deck of 52 cards in a game of 7-card rummy?

Answer: 52C7 = 52! / 7! 45! = 133,784,560

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• Example: How many samples of size n = 6 can be drawn from a population of size N = 50?

Answer: 50C6 = 50! / 6! 44! = 15,890,700

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Boot Camp – End of this module

• This Boot Camp was designed to give you some of the tools you need to successfully work through your Introduction to Statistics course.
• To see the rest of the Boot Camp, visit the full course at http://stats.proffriedman.net/

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