Question 427273: find the value of r if c(18,r)=c(18,r+2)
Found 3 solutions by Theo, robertb, richard1234:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the general formula is:
c(n,r) = n! / (r! * (n-r)!)
the numerator will always be the same.
the denominator also has to be the same.
in order for this to happen, then r! and (n-r)! have to be reversed.
example:
1! * 17! = 17! * 1!
on the left side of the equation, r = 1 and n-r = 17
on the right sides of the equation, r = 17 and n-r = 1
you will get 1! * 17! = 17! * 1! yielding the same denominator.
the difference between the 2, however, is not equal to 2.
when r and n-r get closer together, the difference will be 2.
if we pick 8 as r, then n-r becomes 10.
reversing them, we get r = 10 and n-r = 8.
this satisfies the condition we are looking for so the answer will be r = 8.
with r = 8, the formula becomes c(18,8)
with r = 10, the formula becomes c(18,10)
c(18,8) = 18! / (8! * 10!)
c(18,10) = 18! / (10! * 8!)
the numerator is always the same.
the denominator is the same as well, only commuted.
a*b = b*a is the basic commutative property which is why i say the denominator is commuted.
that's your answer.
r = 8.
Answer by robertb(5830)  (Show Source):
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! A property of combinatorics says that c(N, r) is the same as c(N, N-r). It is easy to prove either algebraically or using a committee-forming argument, since choosing r people on the committee is the same as choosing N-r people *not* on the committee. In addition, C(N, k) = C(N, m) if and only if k+m = N.
Using this property, we can let k = r and m = r+2, and they must add up to 18. Hence,
 -->  --> r = 8.
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