Question

License plates in India begin with a code that identifies the state and district where the vehide 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, ADOO, AD001.... 19999, 80000, 80001, ..., 99999,..., 20000 20001, ...,29999, AA0000, AA0001, .... (a) How many identification numbers are there using two or fewer letters? (b) If a district registers 10 million cars, how many identification numbers must have three letters? (c) Suppose a district registers 406,250 cars. What percentage of identification numbers have exactly one letter? (d) Suppose a district registers 500,000 cars. What percentage of Identification numbers have no letters? (e) Suppose you see a plate in Bangalore, India, with the identification number CR9535. How many vehicles were registered in this district before the vehicle with this plate?

Answer 1

a.7,030,000 license plates

b.2,970,000

c.64%

d.2%

e.969535 plates

Explanation:

(a) We must add the number of possible plates with no letter with ne letter and with 2 letters to account for all using 2 or fewer letters:

10*10*10*10 for the no-letters, ranging from 0 to 9999=10000

26*10*10*10*10 for each of the 26 letters there can be 10000 plates = 260000

26*26*10*10*10*10 for the two letters plates.=6760000.

The sum of the 3 is 7,030,000 license plates.

b) if a state registers 10 million plates, the number of plates with 3 letter will be 10 millions minus the amount of plates that have less than 3 letters, which we calculated previously.

10000000-7030000=2,970,000

c)if a state registers 406,250 cars and we know that one letter plates make 260000 we can use a three rule to detenime the percentage

406,250------100%

260000--------x=26,000,000/406,250= 64%

d)as before we know how many plates can have no letters, so we do a three rule again:

500,000------100%

10,000-------x=1,000,000/500,000=2%

e) We will do this in parts:

From CR0000 to CR9535 there are 9535 cars before (accounting for 0000)

Between CA0000 TO CR0000 there are 17 letters, with 10000 plates each= 170000

Betwen 0000 and CA0000 we have: 10000 for the plates with no letter, 260000 for the plates with one, and 2*260000 for letters A and B followed by another letter: 10000+260000*3=790000

Adding all these numbers: 790000+170000+9535=969535 plates