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

Question

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.

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Answer 1 · 2018-03-16T04:11:56

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 $n=14$