Question 1084494: Solve for r:
C (10,r) = 120
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! c(10,r) = 120
this means 10! / r! = 120
i'm not exactly sure how you would go about solving this except through iteration.
c(10,1) = 10
c(10,2) = 45
c(10,3) = 120
looks like r is equal to 3.
r can also be 7 since the combination formula is symmetrical about the middle.
algebraically, or however you want to call it, c(10,r) is equal to:
10! / (r! * (10-r)!)
what this winds up with is (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) divided by (r! * (10 - r)!
when r = 1, this becomes 10 / 1 = 10
when r = 2, this becomes (10 * 9) / (1 * 2) = 90 / 2 = 45
when r = 3, this becomes (10 * 9 * 8) / (1 * 2 * 3) = 120
when r = 7, this becomes (10 * 9 * 8 * 7 * 6 * 5 * 4) / 1 * 2 * 3 * 4 * 5 * 5 * 7 = 120
you can simplify the arithmetic by using this simple trick.
10! / (3! * 7!) = (10 * 9 * 8 * 7!) / (3 * 2 * 1 * 7!)
the 7! on top and the 7! on bottom cancel out to simplify the operation.
you are left with (10 * 9 * 8) / (1 * 2 * 3) which, with further cancelling becomes (10 * 3 * 4) which is equal to 30 * 4 which is equal to 120.
note that c(10,3) give you the same answer as c(10,7)
c(10,3) = 10! / (3! * 7!)
c(10,7) = 10! / (7! * 3!)
they're the same equation.
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