Arrangements and Combinations Flashcards
Focus on:
- How they have asked the question
- How you can set up an easy solution
There are 5 runners in a race
QA: 10
QB: The total number of possible arrangements for the runners from first to third place
First ask yourself - are these runners coming from one source or several different sources? In this case, they are all coming from one source.
Then ask what is important, in this case what is important is 1st, 2nd and 3rd place so make a mark for those
5 4 3
_ _ _
1 2 3
Then how many different candidates are there for the first spot? How many different people could finish in first? Since there are 5 candidates, the answer is 5
Now, how many candidates are there for the second spot? Well since someone has to be in first, there are 5 - 1 which = 4
Now how many candidates are there for the third spot? Since someone has to be in first and second there are 5 - 2 which = 3
so the answer to the question = B
When you see the word “arrangements” you know that you are doing a permutation, you know that order matters.
When order matters, what do you do?
you multiply!
A congressional committee on legislative procedures is to be made up of four members. If 10 representatives are available, how many different groups are available to make up the four-member committee?
60 120 210 400 720
Are these people coming from multiple sources or a single source?
In this question, they are coming from a single source
Next question, how many people are we selecting? we are trying to make a committee of 4 different people.
10 9 8 7
_ _ _ _
now does order matter? When the GRE asks about “different groups” order does not matter. When order does not matter, you divide by the factorial so
10 9 8 7 (will now be divided by 4! since you are looking for 4 spots) so
10 * 9 * 8 * 7
_ _ _ _
4 * 3 * 2 * 1
Answer = 210
When the GRE asks about “different groups” does order matter?
NO
When order does not matter (like forming different groups or committees) what do you do ?
you divide by the factorial
A co-op board of a certain residential building must consist of two men and three women. If there are six men and seven women available for the committee, how many different groups could be chosen for the committee?
65 525 1,050 1,287 100,800
So start with the men first
there are two spots available for men and a total of 6 men
6 * 5
_ _
2 * 1
No ask, does order matter here? In this question, the answer is no so divide by the number of spaces factorial
7 * 6 * 5
_ _ _
3 * 2 * 1
Then multiply both groups together
15 * 35 = 525
Answer = 525
When are GRE question asks for “different ways”, “different arrangements” or “different groups” what are the steps?
- Determine if there is one source or more than one source
- Draw out slots for each item chosen from the source
- Fill in the slots
- Determine whether the selection order matters
- If there are multiple sources, multiply the results from each source to get the answer
How many sources are there for this situation?
A 4-person committee is to be selected from 6 men and 6 women
1 source
How many sources are there for this situation?
A 4-person committee consisting of 2 men and 2 women is to be selected from 6 available men and 6 available women
2 sources
How many sources are there for this situation?
Trevor wants to purchase 3 books on entrepreneurship. He has found 7 books he’d like to read. How many different groups of 3 books could Trevor buy?
1 source
How many sources are there for this situation?
The 12 students in a class compete in a science fair. How many different ways can the students place first, second and third?
1 source
How many sources are there for this situation?
Kris wants to buy 3 different pairs of gloves, 2 different hats, and 3 different pairs of boots to ride his bike this winter. If he can select from 6 pairs of gloves, 4 hats, and 8 pairs of boots, how many different groups of gloves, hats and boots can he purchase?
3 sources
sundae made with 7 possible toppings
QA: Number of different ways to order a sundae with two toppings
QB: Number of different ways to order a sundae with five toppings
For QA: two toppings so, _ _ and 7 possible toppings so
7 6
_ _
2 1
And because order does not matter, you divide by factorial!
so QA: = 21
For QB: five toppings so, _ _ _ _ _ and 7 possible toppings so start with 7 (just like you did before) and order still matters so divide by factorial
7 6 5 4 3
_ _ _ _ _
5 4 3 2 1
QB: = 21
so answer = c they are equal
When order matters all you do is what?
multiply
