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If $$1216451{\ast}0408832000$$ is equal to $$19!$$, where $$\ast$$ denotes a digit, then the missing digit represented by $$\ast$$ is
$$19!$$ is divisible by 9, so its digit sum must be a multiple of 9. The known digits $$1 + 2 + 1 + 6 + 4 + 5 + 1 + 0 + 4 + 0 + 8 + 8 + 3 + 2 + 0 + 0 + 0$$ add up to 45, already a multiple of 9, so the missing digit is 0 or 9, and only 0 is offered. Indeed $$19! = 121645100408832000$$.
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