What is the smallest value beside 1 of a binomial with two integer values > 0?

Question

I'm searching for the smallest possible value of a binomial(a, b) where a >= b and both values are greater than 0. I calculated a few binomials and always got the result a as the smallest possible value. Is this always the case or just a coincidence?

Answer 1

The $a^\text{th}$ row (the top row is the $0^\text{th}$) of Pascal's triangle contains all of the binomial coefficients of the form $$\binom{a}{b}$$

The smallest non-$1$ element of each row is $$\binom{a}{1}=\binom{a}{a-1}=a$$

Thus, your hypothesis is correct.

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The smallest non-$1$ value over all pairs occurs for $a=2$:

$$\binom{2}{1}=\boxed{2}$$