Section 4 - Council Rock School District



Arrangements and Permutations

Name Period #

Definition 1: A __________________________ of a set of items is an ordered arrangement of the items.

Ex 1: In a SCRABBLE game, Luke drew the letters E, W, L, N, S, F, and O. How many permutations of four of these letters are possible?

ANSWER: There are 7 choices for the first letter, 6 choices for the second letter, 5 choices for the third, 4 choices for the fourth letter. Using what we learned from the Multiplication Counting Principle, there are [pic] or 840 PERMUTATIONS of these 7 letters taken 4 at a time.

* We denote the number of permutations of 7 distinct items taken 4 at a time by: 7P4

* We can also express this product using factorials: 7P4 = [pic]

* Evaluate 7P4 on the calculator:

Evaluate without using the nPr function on your calculator:

2) 11P5 3) 6P4 4) 10P7

5) How many permutations of 6 letters are there from A, E, B, L, N, O, S, T, and Y?

6) What does nPn represent? Explain why nPn = n!

WHAT DO WE DO WHEN AN ITEM IS REPEATED?

Ex 7: When items in a set are repeated, some permutations yield the same arrangement. For example there are 5! permutations of the digits in 11123, but for each permutation, there are 3! ways to permute the 1’s that are indistinguishable. There are only [pic] or 20 distinguishable permutations.

8) How many distinguishable permutations are there of the letters in the phrase “THE EYES”? Can you find an anagram among these permutations?

ANSWER: There are _______________ distinguishable permutations in “THE EYES”.

Any Anagrams?:

9) Give the number of distinguishable permutations of the letters “MISSISSIPPI”.

Permutations Practice Worksheet 1

Name Period #

Examine the letters in the word FAMILY. Fill in the blanks.

1) Consider all 4-letter permuations from these letters, such as FLAM. How many choices are there for the 1st letter? _________. The 2nd letter? _________. 3rd letter? __________. 4th letter? _________

2) Calculate the product of the numbers in a to find the number of 4-letter permutations in the word FAMILY.

________________

3) How many 6-letter permutations are there in FAMILY? ________________

4) Fill in the blank: A ______________________ is an ordered arrangement of a group of items.

Determine whether each statement is TRUE or FALSE.

5) ABC and BAC are permutations of the letters in the word CAB. ________________

6) There are 3 distinguishable permutations of the letters in the name ANN. ________________

Complete.

7) How many permutations of 5 letters are there from A, B, C, D, E, F, G? ________________

8) Give the number of distinguishable permutations in “PHILLIES”. ________________

9) How many ways can a family of 6 line up for a photograph? ________________

Evaluate each expression below.

10) [pic] ________________

11) [pic] ________________

12) [pic] ________________

13) [pic] ________________

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If n and r are positive integers with [pic], then nPr denotes the number of permutations of n distinct items taken r at a time. nPr is given by:

[pic]

If n items include q copies of one item, r copies of another item, s copies of a third item, and so on, then the number of distinguishable permutations of all n items is:

[pic]

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