Question

If 4-digit numbers greater than 5000 are randomly formed from the digits 0,1,3,5 and 7,what is the probability of forming a number divisible by 5 when, a) the digits are repeated? b) the repetition of digits is not allowed?

Answer 1

a) `33/83`

b) `1/4`

Explanation:

a) If the digits can be repeated:

The first digit has to be `5 or 7` to have a number greater than `5000`

The number of possible numbers is: `2xx5xx5xx5 =250`
(`2` choices for the first digit and `5` choices for each of the remaining `3` place holders)

(However this includes the number `5000` which is not greater than `5000`, so there are `249` possible numbers.

Of these, we want the number to be divisible by `5`, so the last digit has to be a `5 or 0`

There are `2xx5xx5xx2 = 100` possible multiples of `5`.
(Remember to exclude the number `5000`, so there are `99`)

`P("divisible by " 5) = 99/249 = 33/83`

b) If the digits may not be repeated.

This means that once a digit has been chosen, there is one digit less for the choice of the next digit.

The number of possible numbers: `2xx4xx3xx2 = 48`
(The possibility of `5000` does not exist this time because `0` can only be used once.)

Of these, we want the number to be divisible by `5`, so the last digit has to be a `5 or 0`

The `5` can be either first or last, but not both.

If the first digit is `5`, then the last digit must be `0`
`color(blue)(5) ? ? color(blue)(0)`

Number of multiples of `5:" "1xx3xx2xx1= 6`

If the last digit is `5` we will have:
`color(blue)(7) ? ? color(blue)(5)`

Number of multiples of `5:" "1xx3xx2xx1= 6`

Therefore there are `6+6=12` possible multiples of `5`

`P("divisible by " 5) = 12/48 = 1/4`