Unlike a permutation, in a combination the order of the obejcts selected does not matter. For example, if you were choosing a team of three students from a class of ten, the order you said “Joe, Sally, John” would not matter to who is in the group- saying “Sally, Joe, John” instead doesn’t change anything.
How to solve a combination
The number of ways you can make a combination of r objects out of a set of n objects is made from the formula below:
You may be asking (like I did when I learned this material), “What? That formula doesn’t make sense!”.- However, it is based on the formula for a permutation. If you think about it, the n! makes perfect sense- thats how many options you’d have if you picked in order from all of them. Removing the (n-r)! gets it down to just the numbers you want (n-r equals the last number you arn’t picking. If you pick 3 from 5, it’ll be 5!/(5-3)!, so 5*4*3*2*1/2*1, which, after cancelling it out to 5*4*3, accomplishes the “pick x of 5″ part.) The extra r! removes all the redundant options. For example, say you have 6 different balls, labeled A through F. If you pick three, you can pick ABC, ABD, ABE, ABF, etc. You’ll end up with six of the same- ABC, ACB, BAC, BCA, CAB, CBA- all of which only count as one in a combination. As you’ll notice, since we picked 3, 3!=3*2*1=6- the same number we have to divide out!
An Example of a Combination
Lets take the example above. Out of a class of ten students, how many ways could you make a team of three students? We know n is ten because the set (the students) has ten objects in it. We know r is 3 because the team will have three students on it. To solve, first we set up the equation:
Then we fill in the values:
By taking apart the factorials, we can simplify the 10! and 7!. Alternativley, just start at 10 and multiply down, stopping right before 7.
There you have it, there are 120 ways to pick a team of 3 people out of ten.
An interesting application of this is the Lottery. See The Lottery to see a good example of it.