Let 5 digit numbers be constructed using the digits 0 , 2 , 3 , 4 , 7 , 9 with repetition allowed, and are…

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, $2\underset{6}{+}\underset{6}{+}\underset{6}{+}\underset{6}{+}=1296$

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

$=1296$

Since, $3\underset{6}{+}\underset{6}{+}\underset{6}{+}\underset{6}{+}=1296$

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

$=216$

Since, $40\underset{6}{+}\underset{6}{+}\underset{6}{+}=216$

And,

$420\underset{6}{+}\underset{6}{+}=36$

$\underline{4} \underline{2} \underline{2}\underset{6}{+}\underset{6}{+}=36$

$423\underset{6}{+}\underset{6}{+}=36$

$424\underset{6}{+}\underset{6}{+}=36$

$427\underset{6}{+}\underset{6}{+}=36$

$429 \underline{0}\underset{6}{+}=6$

$429 2 0=1$

$429 2 2=1$

$429 2 3=1$

Required serial number is

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