Combinations & Permutations

Question 1

Combinations & Permutations

Answer 1

Combinations are used when the order in which a task is completed does not matter.

Permutations are used when the order in which a task is completed matters

Question 2

Combination Formula

Answer 2

C = n!/(k!(n-k!)) —> n choose k

Question 3

The Fundamental Counting Principle

Answer 3

If there are m ways to perform Task 1 and n ways to perform Task 2 and the tasks are independent, then there are m x n ways to perform both of the tasks together.

Question 4

Choosing Multiple Items from Multiple Groups - Using “AND”

Answer 4

With constrains - just MULTIPLY the combinations of each of the groups.

Without constrains - one of the groups will have a constrain, use the combination formula. The other groups’ items will be combined to form a larger group with more items. Use the combination formula on this larger group with choose as the remaining spots.

Question 5

Choosing Multiple Items from Multiple Groups - Using “OR”

Answer 5

With constrains - just ADD the combinations of each of the groups.

Question 6

Choosing “at least” some number of items

Answer 6

Ex: Team of 4 people. At least 2 must be from group A. There are 5 from group A and 4 from group B

Group A (4 choose 2) X Group B (4 choose 2), then ADD,

Group A (4 choose 3) x Group B (4 choose 1), then ADD

Group A (4 choose 4).

The total combinations are the additions of each scenario combo.

Question 7

Collectively Exhaustive Events

Answer 7

If two events are collectively exhaustive. You can use Total Combo = Combo A + Combo B

This can be applied to “at least 1” scenarios. Use Total Combo = At Least 1 + None

Question 8

Dependent Combinations

Answer 8

If the occurrence of event A, changes the occurrence of event B, then apply logic (reduce pool & selection numbers) as you work your way through the events A –> C. Multiply the combinations of all the events and then divide by (#events)!

Question 9

Permutations - The order of the items/objects matters

Answer 9

Permutation formula —> n P k = n!/(n-k)!

Permutation formula for Indistinguishable Items w/o repeats –> N!

Permutation formula for Indistinguishable Items with repeats —> P = N!/[(r1)! x (r2)! x (r3)!…]

Permutation Formula for Circular Arrangements –> (K-1)!

Permutation for items that must be together –> (y-x+1)!(x)!, y is unique items and x are items that must be together.