Question

Given that, C(n,r):C(n,r+1)=1:2 and C(n,r+1):C(n,r+2)=2:3
What is n equals to?​

Answer 1

Step-by-step explanation:

We know that nCr = n!/r!.(n-r)!

nCr/nC(r+1) = 1/2 =>

{n!/r!.(n-r)!}/n!/{(r+1)!(n-r-1)! = 1/2

=> (r+1)/(n-r) = 1/2 => 2r+2 = n-r

=> 3r + 2 = n ……..(1). Also

nC(r+1)/nC(r+2) = 2/3

=>{n!/(r+1)!.(n-r-1)!}/{n!/(r+2)!.(n-r-2)!} = 2/3

=> (r+2)/(n-r-1)= 2/3

=>3r + 6 = 2n - 2r - 2 => 5r + 8 = 2n . Using (1) in this we get

5r+8 = 2(3r+2) => 5r+8 = 6r + 4 => r = 4 using this in (1) we get 3×4+2 = n => n = 14

So r= 4 and n = 14