9.(i)  If \small 4-digit numbers greater than \small 5000 are randomly formed from the digits \small 0,1,3.5 and \small 7, what is the probability of forming a number divisible by \small 5 when, (i) the digits are repeated? 

Answers (1)
H Harsh Kankaria

(i)

Since 4-digit numbers greater than 5000 are to be formed,

The 1000's place digit can be filled up by either 7 or 5 in ^{2}\textrm{C}_{1} ways

Since repetition is allowed, 

Each of the remaining 3 places can be filled by any of the digits 0, 1, 3, 5, or 7 in 5 ways. 

\therefore Total number of 4-digit numbers greater than 5000 = ^{2}\textrm{C}_{1}\times5\times5\times5 -1

= 250 - 1 = 249 (5000 cannot be counted, hence one less)

We know, a number is divisible by 5 if unit’s place digit is either 0 or 5.

\therefore Total number of 4-digit numbers greater than 5000 that are divisible by 5 = ^{2}\textrm{C}_{1}\times5\times5\times^{2}\textrm{C}_{1} -1 = 100 - 1 = 99

Therefore, the required probability = 

P(with\ repetition) = \frac{99}{249} = \frac{33}{83}

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