Question

If $4$-digit numbers greater than $5,000$ 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

(i) When the digits are repeated
Since four-digit numbers greater than 5000 are formed the leftmost digit is either 7 or 5
The remaining 3 places can be filled by any of the digits 0, 1, 3, 5, or 7 as repetition of digits is allowed
$\therefore$ Total number of 4- digit numbers greater than 5000 = 2$\times$5$\times$5$\times$5$\times$=250
A number is divisible by 5 if the digit at its units place is either 0 or 5
$\therefore$ Total number of 4-digit numbers greater than 5000 that are divisible by 5=2$\times$5$\times$5$\times$5$\times$2=100
Thus the probability of forming a number divisible by 5 when the digits are repeated is $\frac{100}{250}=\frac{2}{5}$
(ii) When repetition of digits is not allowed
The thousands place can be filled with either of the two digits 5 or 7
The remaining 3 places can be filled with any of the remaining 4 digits
$\therefore$ Total number of 4-digit numbers greater than 5000=2$\times$4$\times$3$\times$2=48
When the digit at the thousands place is 5 the units place can be filled only with 0 and the tens and hundreds places can be filled with any two of the remaining 3 digits
$\therefore$ Here number of 4-digit numbers starting with 5 and divisible by 5=3$\times$2=6
When the digit at the thousands place is 7 the units place can be filled in two ways (0 or 5) and the tens and hundreds places can be filled with any two of the remaining 3 digits
$\therefore$ Here number of 4-digit numbers starting with 7 and divisible by 5 = 1$\times$2$\times$3$\times$2= 12
$\therefore$ Total number of 4-digit numbers greater than 5000 that are divisible by 5 = 6 + 12=18
Thus the probability of forming a number divisible by 5 when the repetition of digits is not allowed is $\frac{18}{48}=\frac{3}{8}$