Question

Let the number of different car license plates can be constructed if the licenses contain three letters of the English alphabet followed by three digit number
$\left(i\right)$ if repetition is allowed be "k"
$\left(ii\right)$ if repetition is not allowed be "m"
Find [$\frac{k}{m}$] ?(where the brackets imply greatest integer function )

Answer 1

Case (i): Total letters = $26$ and Total digits = $10$
If three letters on plate be represented by alphabets then first place can be filled in $26$ ways,
Second place can again be filled in $26$ ways $(\because$ repetition is allowed)
And third place can also be filled in $26$ ways.
Hence, three letters can be filled by $26\times 26\times 26=(26{)}^3$ ways
And three digit numbers on the plate in $999$ ways $(i.e.001,002,...,999)$
Hence, by principle of multiplication, the required number of ways $=\left(26{)}^3\right(999)$
Case (ii): Here the three letters out of $26$ can be filled in ${}^{26}{P}_3(\because$ repetition is not allowed)
And three digit can be filled out of $10$ in ${}^{10}{P}_3(\because$ repetition is not allowed)
Hence, the required number of ways $=\left({}^{26}{P}_3\right)\left({}^{10}{P}_3\right)$