Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7,...

Number of five-digit numbers formed using the digits $0, 2, 3, 4, 7, 9$ with $2$ at the first place is

$= 6 \times 6 \times 6 \times 6 = 1296$

Since, $2\mathop{+}_{6}\mathop{+}_{6}\mathop{+}_{6}\mathop{+}_{6} = 1296$

Number of five-digit numbers formed using the given digits with $3$ at the first place is

$= 1296$

Since, $3\mathop{+}_{6}\mathop{+}_{6}\mathop{+}_{6}\mathop{+}_{6} = 1296$

Number of five-digit numbers formed using the given digits with $40$ at the first place is

$= 216$

Since, $40\mathop{+}_{6}\mathop{+}_{6}\mathop{+}_{6} = 216$

And,

$420\mathop{+}_{6}\mathop{+}_{6} = 36$

$4 2 2 \mathop{+}_{6}\mathop{+}_{6} = 36$

$423\mathop{+}_{6}\mathop{+}_{6} = 36$

$424\mathop{+}_{6}\mathop{+}_{6} = 36$

$427\mathop{+}_{6}\mathop{+}_{6} = 36$

$429 0 \mathop{+}_{6} = 6$

$42920 = 1$

$42922 = 1$

$42923 = 1$

Required serial number is

$= 1296 \times 2 + 216 + 5 \times 36 + 6 + 1 + 1 + 1 = 2997$