Introduction to Permutations

Show your work for each problem.

1.
What is (3 factorial)?
1. a)27
2. b)3
3. c)6
4. d)9
Answer: 6
2.
Calculate
Answer: 24
3.
What is ?
1. a)Undefined
2. b)1
3. c)0
4. d)Infinity
Answer: 0
By definition, . This is because there is exactly one way to arrange zero objects: do nothing.
4.
In how many ways can 3 people (A, B, C) line up?
Answer: 6
1. How many choices for the first position? 3
2. How many choices for the second position? 2
3. How many choices for the third position? 1
4. Calculate 6
5.
Calculate
Answer: 120
6.
Which formula correctly calculates ?
1. a)
2. b)
3. c)
4. d)
Answer:
, or simply .
7.
Calculate
Answer: 60
, or using the shortcut: .
8.
A club with 7 members needs to elect a President and a Vice President. How many ways can this be done?
Answer: 42
1. How many members can be President? 7
2. After choosing the President, how many can be VP? 6
3. Calculate 42
9.
Which situation requires a permutation (not a combination)?
1. a)Ranking the top 3 finishers in a race
2. b)Choosing 3 pizza toppings
3. c)Selecting 4 students for a committee
4. d)Picking 2 books to read
Answer: Ranking the top 3 finishers in a race
Ranking race finishers requires permutation because 1st, 2nd, and 3rd place are different. Pizza toppings, committee members, and books to read are combinations because order doesn't matter.
10.
How many 3-letter arrangements can be made from the letters A, B, C, D if no letter repeats?
Answer: 24
1. How many letters are available? 4
2. How many letters are we arranging? 3
3. Use . What is ? 24
11.
Calculate
Answer: 56
12.
In a race with 12 runners, how many different ways can 1st, 2nd, and 3rd place be awarded?
Answer: 1320
1. How many runners could finish 1st? 12
2. After 1st place, how many could finish 2nd? 11
3. How many could finish 3rd? 10
4. Calculate 1320
13.
Calculate
Answer: 5040
. This is also equal to .
14.
A 4-digit PIN code uses digits 0-9 with no digit repeated. How many such PINs are possible?
1. a)5040
2. b)720
3. c)10000
4. d)3024
Answer: 5040
PINs. Note: 10000 would be the answer if repetition was allowed.
15.
A class of 20 students needs to select a President, Vice President, Secretary, and Treasurer. How many ways can this be done?
Answer: 116280
1. How many positions need to be filled? 4
2. What permutation formula do we use? P(20,4)
3. Calculate 380
4. Calculate 6840
5. Calculate 116280
16.
Simplify:
Answer: 336
. This equals .
17.
What is equal to for any positive integer ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This makes sense: arranging all objects has ways.
18.
How many 5-letter arrangements can be made from the word STUDY if no letter repeats?
Answer: 120
1. How many unique letters are in STUDY? 5
2. How many letters are we arranging? 5
3. What is ? 5!
4. Calculate 120