Question

A license plate in a certain state consists of 4 digits, not necessarily distinct, and 3 letters, also not necessarily distinct.

(a) How many distinct license plates are possible if no restriction?

(b) How many distinct license plates are possible if it must begin and terminate by a digit?

(c) How many distinct license plates are possible if it must begin and terminate by a letter?

(d) How many distinct license plates are possible if the three letters must appear next to each other?

(e) How many distinct palindrome license plates are possible?
(A palindrome license plate is a license plate that reads the same from left to right as right to left)

Answer 1

There are 10 different digits and 26 different letters.

a)

Each of the four digit can be filled in 10 ways and each of three letters can be filled in 26 ways.

Number of ways of selecting 4 places for digits out of 7 is C(7,4) = 35

The possible number of different license plates is:

35 *10*10*10*10*26*26*26 = 6151600000

(b)

First and last plate can be filled in 10 ways each. Number of ways of selecting 2 more places for digits out of remaining 5 places is C(5,2) = 10.

The possible number of different plates  if it must begin and terminate by a digit is

10 * 10*10*10*10*26*26*26 = 1757600000

(c)

First and last plate can be filled in 26 ways each. Number of ways of selecting 1 more place for letter out of remaining 5 places is C(5,1) = 5.

The possible number of different plates if it must begin and terminate by a letter is

5 * 10*10*10*10*26*26*26 = 878800000

(d)

Assuming 3 letters as one. Number of ways of place for three letters out of 5 is C(5,3) = 10. So possible number of plates is:

5 * 10*10*10*10*26*26*26 = 878800000

(e)

For a palindrome middle place must be filled with a letter. That is middle place can be filled in 26 ways.

Now from starting from first place we have 3 places upto middle, excluding middle place. Number of ways of selecting one place out of three is C(3,1)= 3. Now first three places can be filled in 10*10*26*3 = 7800 ways.

For a palindrome, last three places can be filled in 1 way only.

So possible number of  palindrome license plates is

7800 *26 = 202800