Combinatorics

Question 1

To determine the total number of ways that two or more separate decisions can be made simply multiply the number of ways that each individual decision can be made

Answer 1

4 sizes x 5 colors = 20 different options

Question 2

What does “he selects no more than one from each category” indicate?

Answer 2

There is an additional option of selecting none for each category so add one to each option then apply counting rule

Question 3

What is the counting rule?

Answer 3

To determine the total number of ways that two or more separate decisions can be made simply multiply the number of ways that each individual decision can be made

Question 4

What is a useful technique for handling complex counting problems?

Answer 4

A slot diagram

Question 5

How do you use a slot diagram?

Answer 5

Draw a series of empty slots corresponding to the number of choices that must be made

Fill each slot w the possible number of options for that slot

Apply fundamental counting principle

Question 6

When counting digits what numbers are digits? What number should you not forget?

Answer 6

1-9

Dont forget 0

Question 7

If a code gives you a number, remember to do what when creating your slot diagram?

Answer 7

Put a 1 for that number

Do not put that number in the slot

Question 8

What should you do if your counting problem restricts two or more options from occurring together?

Answer 8

Subtract the number of restricted options from the total number of potential options

Question 9

Remember that a two digit number cannot begin w zero

Answer 9

…

Question 10

What are the two ways to solve an ordering probpem?

Answer 10

Slot diagram

Ordering formula

Question 11

How do you slot an ordering problem?

Answer 11

Write slots, fill them, multiply

Each slot looses an option so its usually in descending order
432*1

Question 12

What is the ordering formula?

Answer 12

n!

N= number of objects being ordered

!= n(n-1)(n-2)(n-3)

Question 13

What is 0!

Answer 13

1

Question 14

How do you work with factorials shown as fractions?

Answer 14

Cancel the largest common element i.e.

8! / 5!3! = 8*7 (canceled 5! )

Question 15

How do you determine the number of unique ways a set of repetitious elements can be arranged?

I.e. how many ways can the letters in the word level be aranged?

Answer 15

Divide the factorial of the total number of elements by the factorial of the number of repetitious elements

I.e. for LEVEL 5!/2!2! = 30

Question 16

What do you do when you have an ordering problem with extra slots?
What if its more than one extra slot?

Answer 16

Consider the slot as an extra object to be ordered

I.e. 5 slots for 4 unique numbers is 5!

Consider empty slots repetitious

Question 17

What is 1!

Answer 17

1

Question 18

What is 6!

Answer 18

720

Question 19

What is 5!

Answer 19

120

Question 20

How do you solve ordering problems that stipulate that certain objects must be together?

Answer 20

Fuse the group that must be together into a single item

Find the new object amount permutation
And multiply that by the permutation of the group that must be together.

I.e. if 8 objects and 3 must be together
The answer is 6! * 3! = 4320

Question 21

How do you solve ordering problems if certain objects cant be together ?

Answer 21

Find the permutation of the number of objects

Multiply the permutation of the group and the restricted objects

Find the difference of the two groups

I.e. a group of 7 objects where 3 cant be together would be

7!

5!3!

7! - (5!3!)

Question 22

Whats the difference between a permutation problem and an ordering problem?

Answer 22

Ordering problems arrange an entire group

Permutation problems arrange a subset of a group

Question 23

Whats the easiest way to solve a permutation problem?

Answer 23

Use a slot diagram

Only create slots for the subset of the objects being asked about

Question 24

What is the formula for circular arrangements?

Answer 24

Total possible linear arrangements/ total spots around the circle

Question 25

A group of elements always has ____ combinations than permutations?

Answer 25

Fewer

Question 26

Whats the difference between a permutation and a combination?

Answer 26

In combinations items are selected and order doesnt matter

In permutation items are arranged and order matters

Question 27

What is the formula for combination problems?

Answer 27

Factorial of the total group/ (factorial of the selected group)(factorial of the unselected group)

Question 28

What are some items that signify a combination problem when the questions asks “how many ways” or “total number of ways”?

Answer 28

Teams, hands, socks
Lines, triangles,diagonals
Toys, tools,trinkets
Colors,labels,tags

Question 29

What type of problem is one where a path can be navigated between two spots on a grid?

Answer 29

Combinatorics

Question 30

Whats the formula for solving a grid combinatorics problem?

Answer 30

Factorial of the total number of blocks/ ( factorial of the horizontal blocks x factorial of the vertical blocks ) are

Question 31

What do you do if you have multiple combinations or arrangements together?

Answer 31

Solve each separately and then multiply their results

Question 32

How do you solve combination questions with “at least”, “or”, “no more than”?

Answer 32

Add their products

You will have to add a successive series of combinations

I.e. all possible combinations of three or more objects , three or less objects

Question 33

What is the formula used to answer a combination problem asking about all the possible combinations?

Answer 33

C=2^n - 1

Where n is the number of total objects

Question 34

“At least one” combination problems are successive combination problems. How do you solve them?

Answer 34

Find the total number of combinations

Then subtract the amount of combinations for the group that is NOT referred to in the “at least” portion of the problem

Question 35

How do you solve combination problems with restrictions ?

Answer 35

Find the must include numbers

And multiply it by the can include combo

I.e. from the word MAGIC how many three letter subgroups can be selected that include the letter A ?

1= the one for A 
6= 4!/(2!2!). For MGIC 
6*1= 6

Question 36

If you have two objects both selected from two different sets . You can calculate the ways of choosing both simultaneously by…

Answer 36

Multiplying them together

I.e. on a menu w 5 apps and 10 entrees there will be 50 different meals

Question 37

You do straight up factorials for…

Answer 37

Ordering n objects

I.e. the number of ways of ordering the letters ABC is 3! Or 6