Mobules 22&23: Combinatorics and Probability

Question 1

Combinatorics vs Permutations

Answer 1

Combinatorics: order doesn’t matter
Permutations: order DOES matter

Question 2

Combination: Basic Combination Formula

Answer 2

nCk = n! / [(n-k)! * k!]

n = total number of objects
k = number of objects that will be chosen

Question 3

Combination: Box & Fill Method

Answer 3

1) Create X number of boxes for the amount of decisions that need to be made. This will be the numerator
2) Fill the box with the number of options available during each choice
3) Fill denominator with the factorial of the number of boxes
4) Solve

Question 4

Fundamental Counting Principal

Answer 4

If there are M ways to perform task 1 and N ways to perform task 2 and the tasks are independent, there are m x n ways to perform both tasks together.

Question 5

AND vs. OR

Answer 5

And = multiply the results
Or = add the results

Question 6

When I see I need to choose “at least” some number of items, I will

Answer 6

probably skip the question…

BUT if I don’t, need to find the combination value for each possible scenario included in the “at least.”

Question 7

Collectively Exhaustive Scenarios

Answer 7

Two events are CE if together they represent all of the potential outcomes. (add all possibilites from each.)

Question 8

When I see a “X items can’t be chosen together” problem, I will

Answer 8

1) Find the total possible ways if there were no restrictions.
2) Find the total possible ways that the two people could be chosen together.
3) Subtract: 1 - 2

Question 9

Basic Permutation Formula

Answer 9

nPk = n! / (n - k)!

n = number of items
k = number of objects chosen

Same as combination without the additional k! multiplied in the denominator.

Question 10

When I see a combination/permutation problem, I will

Answer 10

BEFORE ANYTHING ELSE, determine if the order matters (permutation) or not (combination.) Different strategies, formulas, etc

Question 11

Permutation: Box & Fill Method

Answer 11

1) Create X number of boxes for the amount of decisions that need to be made.
2) Multiply the values in the boxes. That’s it. (Combination requires a denominator of the # of boxes, factorial’d)

Question 12

When I see a permutation problem, I will

Answer 12

be careful not to double-count indistinguishable permutations.

Question 13

Permutation Formula for Indistinguishable Items

Answer 13

N! / r1! x r2! x r3!

Use R as each group of 2 or more indistinguishable items.

Question 14

Permutations: If N Items are to be Arranged in a Circle (Formula)

Answer 14

(N - 1)!

Question 15

When a permutation problem requires 2 items to be adjacent in the order, I will

Answer 15

consider them as one item. Then, to reflect the fact that they could be ordered 1-2 or 2-1, multiply the resulting number of combinations by 2.

Question 16

Basic Probability Formula

Answer 16

number of desirable outcomes / number of total possible outcomes

Question 17

When finding the probability of multiple events occurring, I will

Answer 17

determine if the events are independent (don’t affect the probability of one another) or dependent (they do.)

Question 18

Probability of two non-mutually-exclusive events occuring

Answer 18

P(A) + P(B) - P(Both)

Question 19

If there are X possible ways that an outcome can occur, and each way has the same probability, then the total probability of the situation is:

Answer 19

X * [probability of any one way]

Question 20

When I see “at least” in a probability problem, I will

Answer 20

remember that each number above the “at least” threshold is mutually exclusive, and the probabilities of each need to be ADDED, not multiplied

Question 21

When I See “at least ONE” in a probability problem, I will

Answer 21

use the 1 - x method to calculate the probability faster.

Question 22

Trick to solve algebraic probability problems (probability is given but number of items is not)

Answer 22

Quadratic formula. First scenario, desired value is X, the total value is the total given. The second scenario, desired value is X - 1, total is total - 1.