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?
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xxsl
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You appear to be keeping $a$ fixed and letting $b$ vary. Is that correct? – Tim Raczkowski Jan 15 '15 at 23:53
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a, b both vary. What I'm trying to find out if I can choose a or b within the limitations described above so that a value smaller than a can occur. – xxsl Jan 15 '15 at 23:56
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See Zubin's answer. – Tim Raczkowski Jan 15 '15 at 23:58
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The Question should be edited to reflect the condition stated in the title but not the body of the Question. – hardmath Jan 16 '15 at 00:38
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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.
The smallest non-$1$ value over all pairs occurs for $a=2$:
$$\binom{2}{1}=\boxed{2}$$
Zubin Mukerjee
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