Table of Contents
Definitions
- a Combination is when the order doesn’t matter.
- a Permutation is When the order does matter.
In other words:
A Permutation is an ordered Combination.
Permutations
There are 2 types of permutation:
- Permutation with Repetition: such as the lock. It could be “444”.
- Permutation without Repetition: for example the first three people in a running race. You can’t be first and second.
Permutation with Repetition
The formula is written:
nr
where,
- n is number of things to choose from
- r is number of things we choose of n
- repetition is allowed
- order matters
Permutation without Repetition
where,
- n is number of things to choose from
- r is number of things we choose of n
- repetition is NOT allowed
- order matters
Permutation Examples
1. For my account pin, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 6 of them. (You can choose same number twice or more)
nr = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000
So there are 1,000,000 permutations for my pin.
2. What order could 7 runners be in top 3? (the runner in first place can’t be in second place)
So there are 210 permutations for top 3 runners.
Permutation Notation
Instead of writing the whole formula, people use different notations such as these:
P(n,r) = nPr = nPr =
P(7,3) = 210
Combinations
There are 2 types of combinations (remember the order does not matter now):
- a combination with Repetition: such as coins in your pocket (5,5,10,10,10)
- a combination without Repetition: such as lottery numbers (5,14,17,22,30,34)
Combination with repetition
The formula is written:
where,
- n is number of things to choose from
- r is number of things we choose of n
- repetition is allowed
- order doesn’t matters
Combination without repetition
The formula is written:
where,
- n is number of things to choose from
- r is number of things we choose of n
- repetition is NOT allowed
- order doesn’t matters
Combination Examples
1. There are strawberries, grapes, bananas, pineapples, and apples in refrigerator. I Want to make 3 cups of juice. How many combinations i can make? (repetition is allowed)
So, there are 420 combinations of juices i can make.
2. My teacher tell me to choose 3 girls from 7 girls in my class. How many choices i have? (repetition is not allowed)
Combination Notation
Instead of writing the whole formula, people use different notations such as these:
C(n,r) = nCr = nCr =
C(7,4) =35
Permutation & Combination Problems
1. Grace has 4 skirt, 5 t-shirt, 4 shoes. It can be combined each other. Determine how many ways she can combined skirt, t-shirt and the shoe.
There are:
4 skirt
5 t-shirt
4 shoes
Then 4 x 5 x 4 = 80 ways.
2. Mr. Right’s family will take a photo together. He has two daughters and one son. He wants his child always stand between him and his wife. Determine how many positions to take photo.
There are 3 children that are always between Mr. and Mrs. Right.
So, there is only change the children position 3! = 3.2.1 = 6
Then because of Mr. and Mrs. Right are always in the left – right end position so there are only two possibility (Mr. Right in left side and Mrs. Right in right side or vice versa).
in conclusion, there are 6 x 2 = 12 position.
3. There are 3 boys and 4 girls in a garden. They will watch a show and they sit in a bench. How many ways to arranged them if:
- There is no restriction
- Boys and girls alternate
3 boys
4 girls
- No restriction = 7! = 7P7 = 5040 ways
- Boys and girls alternate = 4 x 3 x 3 x 2 x 2 x 1 x 1 = 144 ways.
Note: red as girls.
4. There are 10 questions in mathematics’ test. Five questions must be done by students. Each student must answer 8 questions. How many ways to choose another five questions?
Total questions 10
5 must be done
There are three questions that must be done and students must choose it from another five questions.
So, it uses combination concept.
5C3 = 10 ways.
5. Sandy is sent to the store to get 5 different snacks of barbeque flavor and 4 different bottles of diet soda. If there are 10 different types of barbeque flavor, and 6 different types of diet soda to choose from, determine how many different Sandy’s choices.
Snacks: 5 from 10 choices
Soda: 4 from 6 choices
Then
10C5 x 6C4 = 252 x 15 = 3780 choices
6. There is a big group of students that consist of 7 men and 8 women. Teacher asked them to make a small group that consist of 5 students and at least 2 men in a group. Determine how many ways can it be done.
Men: 7
Women: 8
Smaller group at least consist of 5 students and at least 2 men.
It can be 2M3W or 3M2W or 4M1W or 5M
Required numbers of ways = (7C2.8C3) + (7C3.8C2) + (7C4.8C1) + (7C5)
= (21.56) + (35.28) + (35.8) + (21)
= 1176 + 980 + 280 + 21
= 2457