An algebra problem by Budi Utomo

Algebra Level 4

\[\small {2016 \choose 0}{2016 \choose 1} + {2016 \choose 1}{2016 \choose 2} + {2016 \choose 2}{2016 \choose 3} + \cdots + {2016 \choose 2015}{2016 \choose 2016} \]

If the sum above can be expressed as \(\displaystyle {\ a \ \choose \ b \ }\) where \(a, b\) are non-negative integers and \(b \le a\), find the smallest possible value of \(a+b\).

Notation: \(\displaystyle { \ M \ \choose \ N \ } = \dfrac {M!}{N!(M-N)!}\) denotes the binomial coefficient.

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