Question 1044455: Could u help me solve this question guys
nCr : nCr-1 = 2:3 , nCr-2 : nCr-1 = 4:3 Find n and r
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! nCr=n!/r!(n-r)!
nC(r-1)=n!/(r-1)!(n-r+1)!, watch the sign there.
nC(r-2)=n!/(r-2)(n-r+2)!
-------------------Divide nCr by nC(r-1)
n!/r!(n-r)!/n!/(r-1)!(n-r+1)! The n! cancels and you invert the denominator
(n-r+1)!(r-1)!/r!(n-r)!
(r-1)!/r!=1/r
(n-r+1)!/(n-r)!=n-r+1
The quotient for this is (n-r+1)/r=2/3
cross-multiply and you get 3n-3r+3=2r
3n+3=5r
------------------------
now do nC(r-2)/nC(r-1)
n!/(r-2)!(n-r+2)!/n!/(n-r+1)!
do the same thing with canceling the n! and inverting the denominator
get (r-1)!(n-r+1)!/(r-2)!(n-r+2)!
this is (r-1)/(n-r+2) =4/3
cross-multiply
3r-3=4n-4r+8
7r=4n+11
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rewrite as
7r-4n=11
5r-3n=3
multiply the top by 3 and the bottom by (-4)
21r-12n=33
-20r+12n=-12
r=21
substitute and n=34
34C21, 34C20, 34 C 19
The first is 927983760, the second is 1391975640, and they are in a 2:3 ratio. Really!
The third is 1855967520
The last divided by the second is 1.3333 repeat, which is 4:3
n is 34
r is 21
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