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**MATHCOUNTS TOOLBOX Facts, Formulas and Tricks**

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Lesson 11: Permutations

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**Consider the possible arrangements of the letters a, b, and c**

Consider the possible arrangements of the letters a, b, and c. List the outcomes in the sample space. If the order is important, then each arrangement is a permutation of the three letters

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**Outcomes in the sample space. abc acb bac bca cab cba **

Consider the possible arrangements of the letters a, b, and c. List the outcomes in the sample space. If the order is important, then each arrangement is a permutation of the three letters Outcomes in the sample space. abc acb bac bca cab cba There are six permutations.

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**Permutation Formula: Use when the item possibilities are all different**

Permutation Formula: Use when the item possibilities are all different. n: number of available items or choices r: the number of items to be selected Sometimes this formula is written: P(n,r).

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**If n = 5 and r = 2, what is the value of 5P2**

If n = 5 and r = 2, what is the value of 5P2? The symbol 5P2 means the number of permutations of 5 items taken 2 at a time.

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**A permutation of 5 objects taken 2 at a time is equal to . . .**

If n = 5 and r = 2, what is the value of 5P2? The symbol 5P2 means the number of permutations of 5 items taken 2 at a time. A permutation of 5 objects taken 2 at a time is equal to . . .

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**A permutation of 5 objects taken 2 at a time is equal to . . .**

If n = 5 and r = 2, what is the value of 5P2? The symbol 5P2 means the number of permutations of 5 items taken 2 at a time. A permutation of 5 objects taken 2 at a time is equal to . . .

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**How many permutations are there for the letters in the word, BANANA**

How many permutations are there for the letters in the word, BANANA? First, notice there are six total letters of which the A is repeated 3 times and the N is repeated 2 times. The number of permutations is equal to

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**How many permutations are there for the letters in the word, BANANA**

How many permutations are there for the letters in the word, BANANA? First, notice there are six total letters of which the A is repeated 3 times and the N is repeated 2 times. The number of permutations is equal to There are six letters in BANANA (6!). The A repeats three times (3!) and the N repeats two times (2!). There are 60 permutations.

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**How many permutations are there for the letters in the word, BANANA**

How many permutations are there for the letters in the word, BANANA? First, notice there are six total letters of which the A is repeated 3 times and the N is repeated 2 times. The number of permutations is equal to There are six letters in BANANA (6!). The A repeats three times (3!) and the N repeats two times (2!). There are 60 permutations.

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**How many permutations are there for the letters in the word, HAPPY**

How many permutations are there for the letters in the word, HAPPY? Remember to ask yourself, how many total letters are there? How many letters are repeated? How many times is each letter repeated?

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**How many permutations are there for the letters in the word, HAPPY**

How many permutations are there for the letters in the word, HAPPY? Remember to ask yourself, how many total letters are there? How many letters are repeated? How many times is each letter repeated? There are five letters in HAPPY (5!). The P repeats two times (2!). There are

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**How many permutations are there for the letters in the word, HAPPY**

How many permutations are there for the letters in the word, HAPPY? Remember to ask yourself, how many total letters are there? How many letters are repeated? How many times is each letter repeated? There are five letters in HAPPY (5!). The P repeats two times (2!). There are

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**1) Calculate the value of 7P3**

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**1) Calculate the value of 7P3**

A permutation of 7 objects taken 3 at a time is equal to . . .

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**1) Calculate the value of 7P3**

A permutation of 7 objects taken 3 at a time is equal to . . .

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**2) Calculate the value of 9P4**

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**2) Calculate the value of 9P4**

A permutation of 9 objects taken 4 at a time is equal to . . .

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**2) Calculate the value of 9P4**

A permutation of 9 objects taken 4 at a time is equal to . . .

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**How many permutations are there for the letters in the word, STATISTICS?**

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**There are ten letters in STATISTICS (10!).**

How many permutations are there for the letters in the word, STATISTICS? There are ten letters in STATISTICS (10!).

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**There are ten letters in STATISTICS (10!).**

How many permutations are there for the letters in the word, STATISTICS? There are ten letters in STATISTICS (10!). The S repeats three times (3!), the T repeats three times (3!), and the I repeats two times (2!).

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**There are ten letters in STATISTICS (10!).**

How many permutations are there for the letters in the word, STATISTICS? There are ten letters in STATISTICS (10!). The S repeats three times (3!), the T repeats three times (3!), and the I repeats two times (2!).

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Fini!

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Multiplication Rule. A tree structure is a useful tool for keeping systematic track of all possibilities in situations in which events happen in order.

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