Introduction to Combinations

Show your work for each problem.

1.
Which formula represents the number of combinations when choosing items from items?
1. a)
2. b)
3. c)
4. d)
Answer:
The combination formula is . The second option is the permutation formula. Options 3 and 4 are for different counting problems.
2.
When should you use combinations instead of permutations?
1. a)When order does not matter
2. b)When items can repeat
3. c)When order matters
4. d)When selecting one item
Answer: When order does not matter
Use combinations when order does not matter. For example, selecting 3 people for a committee - {A,B,C} is the same as {C,B,A}. Use permutations when order matters, like assigning 1st, 2nd, and 3rd place.
3.
Calculate .
Answer: 6
. You can also list them: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}.
4.
Calculate .
Answer: 5
. In general, because there are n ways to choose 1 item.
5.
A bag has 6 different colored marbles. In how many ways can you choose 2 marbles?
Answer: 15
1. What is n (total marbles)? 6
2. What is r (marbles to choose)? 2
3. Calculate . What is the result? 15
6.
Why is ?
1. a)Both equal 21 by coincidence
2. b)The formula has two solutions
3. c)It's a calculation error
4. d)Choosing 3 to include is the same as choosing 4 to exclude
Answer: Choosing 3 to include is the same as choosing 4 to exclude
This is the symmetry property of combinations: . When choosing 3 items from 7 to include, you're simultaneously choosing 4 items to exclude. Both actions represent the same selection.
7.
Calculate .
Answer: 35
8.
A class has 10 students. How many ways can the teacher choose 3 students for a group project?
Answer: 120
ways.
9.
A restaurant offers 9 appetizers. How many ways can a customer choose 4 different appetizers?
Answer: 126
1. What is n (total appetizers)? 9
2. What is r (appetizers to choose)? 4
3. What is ? 3024
4. What is ? 24
5. What is ? 126
10.
A bookshelf has 8 different books. In how many ways can you select 5 books to take on vacation?
Answer: 56
1. What is n (total books)? 8
2. What is r (books to select)? 5
3. Using symmetry, 3
4. Calculate 56
11.
A standard deck has 52 cards. Which expression gives the number of 5-card hands?
1. a)
2. b)
3. c)
4. d)
Answer:
A hand of cards is a selection where order doesn't matter - having {A,K,Q,J,10} is the same hand regardless of the order you received them. So we use possible hands.
12.
Calculate .
Answer: 495
13.
A committee of 5 must be chosen from 8 men and 6 women. How many committees have exactly 3 men and 2 women?
Answer: 840
1. How many ways to choose 3 men from 8? Calculate . 56
2. How many ways to choose 2 women from 6? Calculate . 15
3. Multiply the two results: 840
14.
In a lottery, you choose 6 numbers from 1 to 45. How many different tickets are possible?
Answer: 8145060
1. What is n (total numbers)? 45
2. What is r (numbers to choose)? 6
3. Calculate the numerator: 5864443200
4. What is ? 720
5. Divide: 8145060
15.
If , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
means . Testing: . So . We can verify: .
16.
A student must answer 8 questions out of 12 on an exam. In how many ways can this be done if the first 2 questions are mandatory?
Answer: 210
1. If 2 questions are mandatory, how many more questions must be chosen? 6
2. How many optional questions are there to choose from? 10
3. Calculate . Use symmetry: 4
4. What is ? 210