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C(n,0) = 1

Date: 04/22/2003 at 06:50:38
From: Saurav Kumar
Subject: Combinations

How is C(n,0)=1, where n is an integer?


Date: 04/23/2003 at 09:58:36
From: Doctor Samus
Subject: Re: Combinations

Hi Saurav,

To see why C(n,0) = 1, for n an integer, we must look at the formula 
for C(n,r), when n is an integer, and r is an integer 0 <= r <= n. 
It is:

                n!
    C(n,r) = -------- (where a! = (a)*(a-1)*(a-2)*...*2*1)
             r!(n-r)!

So if r = 0, then we have:

                n!
    C(n,0) = --------
             0!(n-0)!

                n!
           = -------- (since we define 0! = 1)
                n!

           = 1

Another way to think of it is that C(n,r) can be interpreted as the 
number of ways you can select r elements from a set of n elements. 
How many ways can we take 0 elements? Only one: by taking none.

I hope this helps, and feel free to write back if you have any other 
questions.

- Doctor Samus, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Permutations and Combinations

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