If 4-digit numbers greater than 5,000 are ran | vedclass

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Question

(A) When digits are repeated,we form $4$-digit numbers greater than $5,000$ using the set ${0, 1, 3, 5, 7}$.The first digit (thousands place) must be

Vedclass · Accepted Answer

$\frac{33}{83}$

Vedclass · Answer

(A) When digits are repeated,we form $4$-digit numbers greater than $5,000$ using the set ${0, 1, 3, 5, 7}$.The first digit (thousands place) must be $5$ or $7$ to ensure the number is $\ge 5,000$. There are $2$ choices for the first digit.The remaining $3$ places can each be filled by any of the $5$ digits $(0, 1, 3, 5, 7)$ since repetition is allowed.Total $4$-digit numbers formed starting with $5$ or $7$ is $2 	imes 5 	imes 5 	imes 5 = 250$.Since we need numbers strictly greater than $5,000$,we exclude the case $5,000$ itself. Thus,total numbers $= 250 - 1 = 249$.$A$ number is divisible by $5$ if its units digit is $0$ or $5$.For the thousands place,we have $2$ choices ($5$ or $7$).For the hundreds and tens places,we have $5$ choices each.For the units place,we have $2$ choices ($0$ or $5$).Total numbers divisible by $5 = 2 	imes 5 	imes 5 	imes 2 = 100$.Excluding $5,000$ (which is divisible by $5$),the count is $100 - 1 = 99$.The probability is $\frac{99}{249} = \frac{33}{83}$.

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 the digits are repeated?

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