The special factorial inequality

Number Theory Level 3

Consider the inequality below,

\[x! + y! + z! > (x + y + z)!\]

For some ordered triplets \((x,y,z)\), the inequality holds. If \(x,y,z\) ranges all over integer values from \(0\) to \(10\) inclusive, find the numbers of ordered triples that satisfy the inequality.

Details and assumptions

The values of \(x,y,z\) are not necessarily distinct.

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