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I'm trying to figure out some math problems.

In particular I have this "In an office you have 6 clerks. How many ways can you select a team of 3 clerks?" and the solution given is: $$\binom{6}{3} = \frac{6!}{3!*3!} = 20.$$

I don't really understand how that works, so I'm trying to find some tutorial, but how exactly is this equation even called? I can't search for it, because I have no idea what I'm searching :)

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It would help the site if you re-wrote the title of your question to something more specific and descriptive so that it would be helpful later on when someone is searching for something similar to this. –  JohnD Dec 14 '12 at 20:23

2 Answers 2

up vote 6 down vote accepted

What you're referring to is called the "binomial coefficient", spoken "$n$ choose $k$" for natural numbers $n, k$ with $k\le n$.

The equation you posted simply makes explicit how to compute the binomial coefficient of
"$6$ choose $3$," which is denoted $\;\binom{6}{3}$. In this problem, $n = 6, k = 3$.

In general, to compute the number of ways to select $k$ items from $n$ items, without replacement, we compute:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!} = \frac{n(n-1)...(n-k+1)}{k!}.$$


For your problem, we have that $\displaystyle \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac {6!}{3!3!} = \frac{6\cdot 5\cdot 4}{3\cdot2 \cdot 1} = \frac{120}{6} = 20.$

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It's the binomial coefficient, and you should google combination and permutation as a starter.

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Thank you! I will. :) –  NewProger Dec 14 '12 at 18:57

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