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.
∴Total number of 4-digit numbers greater than $5000 = 2 × 5 × 5 × 5 - 1$
$= 250 - 1 = 249$
[In this case, $5000$can not be counted; so $1$ is subtracted]
A number is divisible by $5$ if the digit at its unit place is either $0$ or $5$. ∴Total number of 4-digit numbers greater than 5000 that are divisible by 5 $= 2 × 5 × 5 × 5 - 1 = 100 - 1 = 99$
Thus, the probability of forming a number divisible by $5$5 when the digits are repeated is $=\dfrac{99}{249}=\dfrac{33}{83}$

(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.
∴Total number of 4-digit numbers greater than $5000 = 2 × 4 × 3 × 2 = 48$
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.
∴Here, the number of 4-digit numbers starting with 5 and divisible by $5 = 3 × 2 = 6$ When the digit at the thousands place is 7, the unit 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.
∴ Here, the number of 4-digit numbers starting with 7 and divisible by $5 = 1 × 2 × 3 × 2 = 12$5 $=$ $1 × 2 × 3 × 2 = 12$
∴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 $\dfrac{18}{48}=\dfrac{3}{8}$