1. Lesson # 65 Notes Combinations

2. Lesson # 65 Combinations

3. Combinations - Combinations are a mixture of the Counting Principal & Permutations. The number of combinations is a fraction of the number of permutations. The order of the items chosen does not matter. Combinations are groups of things. Combinations are a mixture of the Counting Principal & Permutations. The number of combinations is a fraction of the number of permutations. The order of the items chosen does not matter. Combinations are groups of things. Follow these steps to solve combinations: Follow these steps to solve combinations: 1. Set up and solve for the permutation. (decide how many spots 1. Set up and solve for the permutation. (decide how many spots to fill or pick) 2. Start with your biggest number & count down until you run 2. Start with your biggest number & count down until you run out of spots. The permutation you solved is the top of your fraction. The next step (the factorial) is the bottom of your fraction. 3. Find the factorial of the number of spots you have to fill or pick. (multiply down to 1) 3. Find the factorial of the number of spots you have to fill or pick. (multiply down to 1) 4. Divide the top row (the permutation) by the bottom row. 4. Divide the top row (the permutation) by the bottom row. (your factorial)

4. Example: How many groups of 3 songs can Mary choose from a CD containing 12 songs? 12 C 3 = 12 P 3 = 12 x 11 x 10 = 1320 = 220 ways 3 x 2 x 1 6 3!

5. Try this combination Try this combination

6. Step # 1 Write or start with the formula - 9 C 4 = 9 C 4 =

7. Step # 2 Next - find the number of permutations - 9 C 4 = 9 P 4 = 9 x 8 x 7 x 6 = 3024 9 C 4 = 9 P 4 = 9 x 8 x 7 x 6 = 3024

8. Step # 3 Find the factorial of the number chosen or picked. Write it under the permutation. 9 C 4 = 9 P 4 = 9 x 8 x 7 x 6 = 3024 9 C 4 = 9 P 4 = 9 x 8 x 7 x 6 = 3024 4!4 x 3 x 2 x 124

9. Step # 4 Simplify the fraction to get the number of combinations. 9 C 4 = 9 P 4 = 9 x 8 x 7 x 6 = 3024 = 9 C 4 = 9 P 4 = 9 x 8 x 7 x 6 = 3024 = 4!4 x 3 x 2 x 124 126 ways

10. Permutation vs. Combination Pick 3 songs out of 5 songs. Permutation - (order matters) Permutation - (order matters) 5 P 3 = 5 x 4 x 3 = 60 ways to play them. 5 P 3 = 5 x 4 x 3 = 60 ways to play them. Combinations – (order doesn’t matter) Combinations – (order doesn’t matter) 5 C 3 = 5 P 3 = 5 x 4 x 3 = 60 = 10 groups of 3 songs. 3!3 x 2 x 16