Imagine that you are the leader of a ninja clan. You have 10 ninjas at your disposal, with code names A, B, C, D, E, F, G, H, I, and J. Let's say that you need 6 of them for a special secret ninja mission. How many ways can you choose the 6 ninjas?
First, notice that the 6 ninjas don't need to be put in any specific order, nor do they need to be assigned 6 distinct jobs in the mission. You just need to choose 6 out of 10 of them, period--there is no way to distinguish ABCDEF from FEDCBA. Because order does not matter, a mathematician would say that you're looking for the number of 6-ninja combinations (as opposed to permutations).
To find the number of combinations, start by finding the number of permutations. If you don't know how to do this, read the article Calculating Permutations. You will obtain the answer 151,200. This represents the number of ways you can choose 6 ninjas, in a certain order, out of a pool of 10 ninjas.
But this number is too big. You have counted ABCDEF, ABCDFE, and FEDCBA all as separate possibilities, when they're really the same. What you need to do is to divide by the number of times you repeated each choice. Ask yourself,
"In how many ways can I arrange ABCDEF?" Using your strategies for permutations, you will find that there are
6*5*4*3*2*1
= 720 ways.
In short, for every combination of 6 ninjas, there are 720 permutations. You know the number of permutations already, so simply divide that by 720. The number of combinations is
151,200/720
= 210.
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To generalize: to find the number of combinations of n (10 in our example) things chosen k (6 in our example) at a time,
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Written with factorials, that's
