Question

A moped licence plate has two letters and then four numbers in it. How many plates can be made without duplicating, there are no plates with number zero?

Answer 1

Hint: We should find all the possible ways of combination where there is no duplication and there are no zeros. So first we should find out all possible combinations with 2 letters and 4 digits. Then we will subtract that value with the total number of ways where there are2 letters and zeros.

Complete step-by-step answer:
We know that in the licence plate the first two are letters let us consider them as $ {L_1} $ and $ {L_2} $ and the next four are digits let us assume them to be $ {N_1},{N_2},{N_3}and{N_4} $ (these digits are non-zero)
Hence the licence format is $ {L_1}{L_2}{N_1}{N_2}{N_3}{N_4} $
Number of ways to select letter is 26 because there are 26 alphabets in English
Number of ways to select numbers is 9 (including 0)
Hence the total number of ways of selection including zero are
 $ = 26 \times 26 \times 10 \times 10 \times 10 \times 10 $
=6760000
Now, the number of ways of selecting zero is
 $ = 26 \times 26 \times 1 \times 1 \times 1 \times 1 $
=676
Hence, the total number of ways in which plates can be made without duplicating and no plates with number zero are
 $
   = 676000 - 676 \\
   = 675324 \;
  $
So, the correct answer is “675324”.

Note: In this particular question the order doesn’t matter hence we didn’t use permutation. We only used combinations to solve this problem. A combination is basically the selection of an object irrespective of the order.