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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?

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Solution

(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×=250
A number is divisible by 5 if the digit at its units place is either 0 or 5
Total number of 4-digit numbers greater than 5000 that are divisible by 5=2×5×5×5×2=100
Thus the probability of forming a number divisible by 5 when the digits are repeated is 100250=25
(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 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 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
Here number of 4-digit numbers starting with 7 and divisible by 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 1848=38

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