Let $n$ an $8$ -digit integer number formed by six different digit and begins with number $5$ . Determine how many $n$ If it must contains three digit $t\ne5$ .

*
Note:
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Example of
$n$
is
$51223426, 50413211, etc.$

The answer is 529200.

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There are $9$ ways to choose $t$ , then choose $3$ position out of $7$ available position to put the number $t$ in the $8$ -digit number $C_3^7$ , lastly we will choose $4$ other digit to fill the empty position in the $8$ -digit number out of $8$ available number and then arrange them $C_4^8\cdot 4!$ . $9\cdot C_3^7\cdot C_4^8\cdot4!=529200$