Combination and Permutations

Question 1

Combination

Answer 1

the order does not matter

Question 2

Permutation

Answer 2

the order does matter

Question 3

Combination example

Answer 3

• select 5 candies from a dish containing 10 different color candies
• A tycoon has 10 different sports cars in her garage and she will choose three of them to wax. How many different sets of three cars could she choose?

Question 4

Permutation example

Answer 4

A tycoon has 10 different sports cars in his garage. How many possible ways are there for him to arrange three of the cars in a line?
- It will matter which car goes where

Question 5

Combination Formula

Answer 5

nCk= n!/ (k!(n-k)!)

Question 6

Box and fill method for Combination problem

Answer 6

let each box represent a specific decision that must be made. Then divide the product of all the numbers in the boxes by the factorial of the number of boxes that have numbers inside.

A coach must select 3 softball players from a pool of 7 possible players to play for his team. How many different ways are available to the coach in selecting these 3 players?

3!
= 35

Question 7

If there are m ways to perform one task and n ways to perform another task and of the tasks are independent

Answer 7

m*n possible ways to perform another task

i. e We are making breakfast one morning and that we can choose among 5 types of eggs, 3 types of coffees, and 6 types of pastries. If a breakfast consisting of 1 egg, 1 pastry, and 1 types of coffee, we can then prepare breakfast in one of a total of 536= 90 different ways
i. e: If nickel is flipped five times, how many different head and tail sequences are possible?
- > 2^5= 32

Question 8

Choosing multiples items from the word “or”

Answer 8

we’re being asked to choose between alternatives

Question 9

Mutually exclusive

Answer 9

If two event cannot occur together

-> stock market can not go up and down and be unchanged -> not all can happen at the same time

Question 10

Mutually exclusive -> ways to accomplish it

Answer 10

If there are x ways to accomplish event A and y ways to accomplish event B and if A and B are mutually exclusive then there are x+y ways to accomplish A or B

Question 11

Collectively exhaustive

Answer 11

those events represent all of the possible outcomes of a scenario

Question 12

The total ways in which collectively exhaustive scenario can occur

Answer 12

The number of ways in which A can occur + number of ways in which b can occur

Question 13

Some items can never be together in the same group

The total 3 person club configuration that can be made

Answer 13

number of 3 person club configuration in which both people are together in the club + number of 3 person club configuration in which both people are not together in the club

Question 14

three people are to be selected from a pool of five people to form a writing club. If two of the people can never be together in the club, in how many different ways can the club be formed?

Answer 14

No restrictions
5C3= 10
Two people are together
3C1= 3

10-3= 7 people

Question 15

number of 3 person club configuration in which both people are not together in the club

Answer 15

The total 3 person club configuration that can be made - number of 3 person club configuration in which both people are together in the club

Question 16

number of ways in which at least one of the x items must be chosen from y items

Answer 16

total number of ways to choose x items from y items - the number of ways to choose none of the x items from y items

Question 17

An international rugby team composed of four people must be organized out of a pool of five players from turkey and six players from Iran, If a team must have at least one player from Iran, in how many different ways can the team be selected?

Answer 17

total # of ways to form team= total- none

11C4= 330
5C4= 5
= 330-5

Question 18

Dependent Combinations

Answer 18

Two events A and B are dependent if the number of ways in which event B could be selcted depends on which specific way even A is selected

Question 19

Calculate an unknown number of items in a group

Answer 19

set up a box and fill diagram

->

Question 20

Permutation

Answer 20

the order of the selection of objects in which the orders of the objects matters

-> In a competition involving ten competitors, one Gold, one sliver and one bronze medal will be awarded. How many different four digit number sequences are possible ?

Question 21

Permutation formula

Answer 21

nPk= n!/ (n-k)!

Question 22

Permutation formula using the box method

Answer 22

let each box represent a specific choice that must be made. Multiply the number in each box -> you don’t need to divide by the factorials in the denominator unlike for combination

Question 23

indistinguishable items in permutation problems

Answer 23

count only the number of distinguishable permutations

-> How many ways are there to arrange SSSSS -> only one way

Question 24

Permutation formula for indistinguishable items

Answer 24

P= n!/ (r1!)*(r2)!…(rn!)

-> How many ways can A,A,B,B be arranged
P = 4!/ (2!*2!)= 6 ways

Question 25

Circular arrangement formula

Answer 25

(k-1)!

Question 26

Permutation anchoring to solve restrictions

Answer 26

first, place the restricted items into their specific spots in the arrangement then handle the additional items that can be arranged in the remaining spots

Question 27

When some items must be together

Answer 27

link those items together

Question 28

When two items cannot be next to each other

Answer 28

Two outcomes
1) WHere they stand next to each other
2) When do not stand together
= Total - when they are together