Have you ever wanted to calculate the number of possible ways to choose a select amount of items from a group of items? This lesson will teach you about the basics of using combinations.
ISAIAH LOPEZ
Combinations represent choosing a certain amount of items from a certain amount of items. In order to use combinations, you need to understand factorials. If you don't, check out the Four Cells article on that. There is a basic formula for combinations. It is written below.
The parenthesis with the "n" and "r" in it is the way we write combinations. They can also be written like this: nCr (for "n choose r"). n represents the amount of items we have to choose from, and c is the number of items we are choosing. For example, if we are choosing 3 people at random out of a group of 7, we can write 7C3, or "7 choose 3". The final answer will give you the number of possible ways there are to do this.
Now lets get into the formula. The formula for choosing r items from a group of n, as listed above, is n factorial divided by (r) factorial times (n - r) factorial. So if we are choosing 3 items from a group of 7, the answer would be 7! divided by 3!(7 - 3)!, which simplifies to 35. Lets try another one! If we are choosing 4 items from a group of 6, the number of possible ways to do this would be 6! divided by 4!(6 - 4)!, which simplifies to 15. One thing to note is that this number will always be a positive integer, because the number of ways to do something will always be a positive integer.
Once these numbers get large, the math become a lot more complicated and tedious. This can take a lot longer than it should. This is why there is a faster and easier way to find the number of combinations. It involves using permutations. If you don't know what permutations are, I recommend you check the article on that, but you will still be able to follow along. The faster formula is (nPr) divided by (r!), which is expressed in terms of a permutation. If you want to do it intuitively, just take the first r terms of n factorial and divide them by r!. For example, in 6C4, you can take (6 · 5 · 4 · 3) and divide it by (4!). Another example is, in 8C5, take (8 · 7 · 6 · 5 · 4) and divide it by (5!). I hope this article helped you understand combinations.