Question

License plates in India begin with a code that identifies the state and district where the vehicle is registered, and this code is followed by a four-digit identification number. These identification numbers are given sequentially, starting with 0000, 0001, 0002, etc. Once this sequence reaches 9999, a letter from the set (A, ..., Z) is added (in order), and once these run out, additional letters are added, and so on. So the sequence of identification numbers proceeds as follows: 0000, 0001,, 9999, A0000, A0001, .., A9999, B0000, B0001, ..., B9999,, z0000, Z0001,, Z9999, AA0000, AA0001, .. How many identification numbers are there using three or fewer letters? (b) If a district registers 200 million cars, how many identification numbers must have four letters?

Answer 1

a: 175,760,000

b: 24,240,000

Explanation:

For each letter assigned, there can be 10,000 numbers after it (the numbers 0000 through 9999).

There are 26 letters, so when we use all the letters we have assigned

26*10,000 = 260,000 license plates.

Each time we add a letter, we have assigned 26 more options

(AA, AB, ... AZ, BA, BB, ...BZ)

So when we get through 2 letters we have

26*260,000 = 6,760,000 plates

When we add a third letter, we have 26 new cycles of the previous situation.

So we have 26*6,760,000 = 175,760,000 plates

If the district has 200,000,000 cars, then they need

200,000,000 - 175,760,000 = 24,240,000 cars with 4 letters on their plates