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
(i) the digits are repeated?
(ii) the repetition of digits is not allowed?

Answer 1

A $4$ digit number greater than $5000$ is randomly formed from digits $0,1,3,5,7$.

(1) Repetition is allowed:

We need to form a number greater than $5000$, hence, the leftmost digit can be either $5$ or $7$.

Since repetition of digits is allowed, so the remaining three places can be filled by $0,1,3,5,or7$.

Hence, the total number of $4$ digit numbers that can be formed greater than $5000$ are = $2\times 5\times 5\times 5=250$

But, we can’t count $5000$ so the total number becomes $250–1=249$.

The number is divisible by $5$ only if the number at unit’s place is either $0or5$.

Hence, the total number of numbers greater than $5000$ and divisible by $5$ are = $2\times 5\times 5\times 2$ – 1 = 99

Hence, the required probability is given by = $\frac{99}{249}$ = $\frac{33}{83}$.

(2) If repetition of digits is not allowed:

For a number to be greter than $5000$, the digit at thousand’s place can be either 5 or 7.

The remaining three places can be filled by any of the four digits.

hence, total number of numbers greater than $5000=$ $2\times 4\times 3\times 2=48$.

When the digit at thousand’s place is 5, units digit can be 0 and the tens and hundreds digit can be any two of the remaining three digits.

Hence, the number of 4 digit numbers starting with $5$ and divisible by $5=$ $3\times 2=6$

When the digit at thousand’s place is 7, units digit can be filled in two ways (0 or 5) and the tens and hundreds digit can be any two of the remaining three digits.

Hence, the number of 4 digit numbers starting with 7 and divisible by 5 = $1\times 2\times 3\times 2=12$.

therefore, the number of 4 digit numbers greater than 5000 and divisible by $5=12+6=18$

hence, the required probability = $\frac{18}{48}=\frac{3}{8}$