# 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?

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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