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I discovered following identity via a programming exercise. Looking for the way how it prove it by induction or some other way.

$$\binom{a_1+a_2+\cdots+a_t}{a_1,a_2,\cdots,a_t}=1+\sum_{i=2}^t \sum_{j=1}^{i-1} \sum_{k=1}^{a_i} \binom{ a_1+a_2+\cdots+a_{i-1}+k }{a_1,a_2,\cdots,a_j-1,\cdots,a_{i-1},k }$$

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