Math question with binomials?

Question

Find three consecutive binomial coefficients in the ratio 1 to 2 to 3. That is,
(k):(k + 1):(k + 2) = 1:2:3

Answer 1

To find these, start by setting C(n,r)=1/2*(C(n,r+1)).
n!/[(n-r)!(r!)]=
1/2*n!/[(n-(r+1))!*(r+1)!]

This simplifies to
(n-[r+1])!*(r+1)!/([n-r]!*r!)=1/2,
or (r+1)/(n-r)=1/2.

This becomes 2(r+1)=n-r or n=3r+2.

Also, set C(n,r+1)=2/3*C(n,r+2).

Solving this gives you another equation with n and r.
Solve the two equations with n and r together.

This gives you n=14 and r=4, and 1001, 2002, and 3003 as the numbers you seek.