1. Permutations and Combinations Section 2.2 & 2.3 Finite Math

2. Bellwork Compute 1. 5P3 2. 7P2 3. 10P6 4. 8P5 5. 4P2

3. The Letter Permutation Problem Lets consider the word “infinite.” How may 8-letter arrangements can be formed from the letters of this word? This is a special problem that requires its own trick. Here’s how it goes: First, treat it as a normal 8-slot permutation: ◦ 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320

4. The Letter Permutation Problem However, that’s too many, because some letters are repeated. So, we divide it by the factorial of number of times a letter is repeated. For example, “i” is repeated 3 times, so you divide by 3!. “n” is repeated twice, so divide by 2!: ◦ 40320/3!2! ◦ Therefore, the answer is 40320/(6*2) = 3360.

5. Practice Problem How many 6 letter arrangements can be formed from the word “effect”? a) 30 b) 60 c) 180 Answer: C

6. Combination Combination is just permutation – you are counting the number of ways to pick from a set without repetition of elements The difference is that, for combination, order does not matter Slot method can only be used when order matters; therefore, you cannot use slots for a combination.

7. Combination Formula: ◦ C(n,r) = n!/(n-r)!r!  n = number of things you are choosing from  r = number of things you are choosing Since slots cannot be used, this formula is your only tool in solving combination problems

8. Combination Let’s do an example combination problem. You order Mother Bear’s Pizza with a Pizza with a friend late at night. There is a special o a 3-topping pizza so you decide to go with that. There are 8 toppings to choose from. How many different pizzas can possible be made?

9. Combination Simply apply the formula here. n = 8 (8 toppings to choose from), r = 3 (3 toppings being chosen to put on the pizza): ◦ C(8,3) = 8!/(8-3)!3! ◦ Which comes to 56

10. Distinguishing between Permutation and Combination Permutation = order matters Combination = order does not matter Let’s think of it another way-if the “slots “ are distinguishable between each other, the order matters (P). I other words, if you rearrange the same elements, it becomes a different set. Ex: A 3-digit number from {1,2,3,4,5}. This is a permutation, since it digit 1 and digit 2 are “distinguishable.” If you switch them, it becomes a different number. 123 is different than 321

11. Distinguishing between Permutation and Combination On the other hand, if the slots are used “indistinguishable,” than order doesn’t matter (C). In other words, if you rearrange the same elements, it is still the same thing. Ex: 3 toppings from 10 on a pizza. You cannot distinguish between topping 1 and topping 2; Pepperoni, Sausage and Ham is the same thing as Ham, Pepperoni, and Sausage. Remember, neither (P) or (C) can have repetitions. If there are repetitions, use neither of these two methods

12. Practice Problem Identify whether each problem is a permutation, combination, or either. 1. Number of ways to form a committee of president, VP, and treasurer from 10 students 1.Permutation 2. Number of ways to select 5 distinct roles for a play out of 10 potential actors 1.Permutation 3. Number of ways to pick a hand of 5 cards from a deck of cards. 1.Combination