GMAT Quant Chapter 15: Combinations & Permutations

Question 1

What question can we ask to determine whether a combination of elements question requires a combination or a permutation of choices?

Answer 1

Does the order of the items matter? (If one item occurs once can it never occur again in that set of choices?)

order does not matter for combinations

order does matter for permutations

Question 2

What is the basic combination formula?

Answer 2

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

n is number of objects from which we’ll choose
k is number of objects we actually choose
n choose k

Question 3

What is a quick visualization tool to implement the basic combination formula?

use 5 choose 3 to demonstrate

Answer 3

The Box & Fill

5 x 4 x 3
/
3!

Question 4

In complex combination problems, the words “and” and “or” trigger us to do what?

Answer 4

And means multiply the combinations
Or means add the combinations

Question 5

Define mutually exclusive events.

Answer 5

Mutually exclusive events cannot occur simultaneously.

Question 6

Typically, at least combination problems require extra calculations because?

Answer 6

When we solve an “at least” combination problem we need to calculate all of the individual combinations in included in the at least set and then add them up.

Question 7

How do we know if a combination question is a restriction combination problem?

How do we solve restriction combination problems when items must be included in the subgroup?

Answer 7

If we’re told that certain items MUST be selected then we have a restriction combination problem

Subtract the items that must be included in the subgroup from the main group & subgroup.
Then perform normal nCk operation with smaller choice options.

Question 8

How do we solve restriction combination problems when some items must not be chosen?

Answer 8

We subtract the items that must not be included from the total number of items in the main group.

The number of items to be selected in the subgroup is unchanged.

Question 9

How do we solve a combination restriction problem wherein some items must be chosen and some items must not be chosen?

Answer 9

Subtract the items that must be chosen from the subgroup and main group

Subtract the items that can’t be chosen from the main group.

Question 10

Define collectively exhaustive.

Answer 10

Events that represent the total range of possible outcomes.

Question 11

What 2 facts, if true, can save us time when calculating “at least” combination problems?

How can we use these 2 facts to calculate the number of possibilities for a specific type of event?

Answer 11

1. Collectively exhaustive
2. Mutually exclusive

The total number of possibilities = the number of ways in which A can occur + the number of ways in which B can occur.

Question 12

How can we calculate the number of possibilities when two items cannot be together in a group?

Answer 12

1. Calculate the total number of possibilities
2. Calculate the possible combinations of the two items together.

Subtract 2 from 1

Question 13

How can we use the facts of collectively exhaustive and mutually exclusive events to calculate the number of possible ways at least one item can be selected from a group?

Answer 13

Determine all possible combinations

Subtract out the smaller number of possibilities for the subgroup that we don’t want

Question 14

What is the basic permutation formula?

Answer 14

nPk = n! / (n-k)!

n is the number of objects from which a choice can be made
k is the number of objects that are to be chosen

Question 15

How is the box and fill method different for permutations?

Answer 15

We only need to multiply the boxes by each other,

e.g. 6!

Question 16

How does the presence of indistinguishable items change the permutation formula?

What effect does their presence have on the number of total possibilities?

Answer 16

Number of possibilities =
n!
/
frequency of indistinguishable object one!
*
frequency of indistinguishable object two!

It reduces it.

Question 17

How does the number of possibilities change when the items are arranged in a circle rather than a line?

What is the formula for arranging items in a circle?

Answer 17

The number of possibilities in a circular arrangement is smaller than a line arrangement.

(k-1)!
k is the number of items to be arranged in a circle

Question 18

What is the best technique to solve permutation problems with restrictions?

Answer 18

Anchor method.

Keep restricted items in their specific positions

e.g. because their position is fixed, you only need to order all the other items in the set

so if the problem was 6 items with 1 item 2nd, the solution is 5!

Question 19

How do we solve permutation problems when items must be linked together?

What must we remember with linked permutation problems?

Answer 19

Think of the linked item as one large item.

If we have a group of 8 items with 3 linked. Instead of 8! the solution is 6! because 5 for the normal items and 1 for the group of linked items

If the items are linked but different, there are multiple possibilities based on the order of the items in the link, George in front of Stacey is different to vice versa.

Thus multiply the number of possibilities by the number of items in the link that can be arranged differently.

Question 20

How do we solve a permutation problem when more than one item needs to be fixed in a certain place?

e.g. 6 people in a line with George always second and Stacey always fifth

Answer 20

Exactly the same way we solve them when one person is fixed. Anchor method.

Fill the boxes with the items in their specific places and the solution is the factorial of the number of remaining spots to be chosen.

Question 21

How do we solve a permutation problem when two items cannot be together?

Answer 21

Use the fact that the events are mutually exclusive and collectively exhaustive.

Determine the total number of possibilities and subtract the number of possibilities of the outcome that we don’t want.