Question

How do you find three consecutive binomial coefficients in the relationship `1:2:3`?

They are `((14),(4))`, `((14),(5))`, and `((14),(6))`, but I'd like to know how to obtain that result without using Pascal's Triangle.

Answer 1

The ratio of two consecutive binomial coefficients is given by :

`(((n),(r+1)))/(((n),(r)))= (n!)/((r+1)!(n-r-1)!) times (r!(n-r)!)/(n!) = (n-r)/(r+1)`

So, for `((n),(r)), ((n),(r+1))`, and `((n),(r+2))` to be in the ratio 1:2:3, we must have

`(n-r)/(r+1) = 2 implies n =3r+2`

and

`(n-r-1)/(r+2)=3/2 implies n = 5/2r+4`

The two relations together give

`3r+2=5/2r+4 implies r/2=2 implies r=4`.

Using either of the two relations then leads to `n=14`