[This post is targeted at a Level 2 student. You should be familiar with Permutations II.]
A combination is a way of choosing elements from a set where order does not matter. Consider the following question: A high school physical education teacher has to decide on which two sports the students will be able to play in class today. There are 6 different sports that the school has the correct equipment for. How many choices for the two sports does the coach have?
Recall that in permutations, we said that if we wanted to arrange 2 items from 6 elements, there would be ways that we could do that. This doesn’t quite solve our problem, since the teacher sees no difference from choosing soccer first and then basketball, or choosing basketball first and then soccer. So each possible choice is counted twice, and we will have to divide our answer by 2. So the answer is actually .
We can generalize this idea using the following example: Lisa has 12 different ornaments in her closet. She wants to give five to her mom as a birthday gift. How many ways can she do this?
We can again think of giving Lisa’s mom a first ornament, a second ornament, a third ornament, etc. So we know that this would give choices. Lisa’s mom is receiving all five ornaments at once, so the order in which we decide on them does not matter. If we pick the bear first and then the frog second, it’s the same as if we picked the frog first and the bear second. How many ways could we have picked any set of 5 ornaments? There are orderings of the chosen ornaments, so we could have picked a certain combination different ways. Thus the total number of choices for Lisa is .
Notice that in the answer, the factorials in the denominator sum to the value in the numerator. This is not a coincidence. In general, if we are picking elements from an element set and we do not care about the order, there are ways to do it. We represent this as a binomial coefficient, using the notation .
1. How many ways are there to arrange 3 chocolate chip cookies and 10 raspberry cheesecake cookies in a row of 13 cookies?
2. How many ordered non-negative integer solutions are there to ?
For an extremely detailed explanation, you should refer to Solution to a Combinatorics Challenge.
3. Both of the previous answers are 286. Is it a happy coincidence?
1. How many ways can three different appetizers be chosen from a menu that has 10 choices?
2. In New York city, all the streets are arranged in a grid. A hospital is located 5 blocks east and 6 blocks north of an accident. How many ways are there to get there, if we only go 1 block north or 1 block east at each intersection?
3. An office with 8 people wants to have 2 teams of 2 players to participate in a charity tournament. How many ways can the teams be made? Note: A player cannot be in multiple teams.
4. How many ordered integer solutions are there to subject to ?
5. Winston must choose 4 courses for his final semester of school. He must take at least 1 science class and at least 1 arts class. If his school offers 4 science classes, 3 arts classes and 3 other classes, how many different choices for classes does he have?