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(B) number between $2000$ and $5000$ is a $4$-digit number. For a number to be a multiple of $3$,the sum of its digits must be divisible by $3$. We se

Vedclass · Accepted Answer

$30$

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(B) number between $2000$ and $5000$ is a $4$-digit number. For a number to be a multiple of $3$,the sum of its digits must be divisible by $3$. We select $4$ digits from ${0, 1, 2, 3, 4}$ such that their sum is divisible by $3$: $1)$ ${0, 1, 2, 3}$,sum $= 6$ (divisible by $3$). $2)$ ${0, 2, 3, 4}$,sum $= 9$ (divisible by $3$). For the set ${0, 1, 2, 3}$,the thousands place can be $2$ or $3$ ($2$ choices). The remaining $3$ positions can be filled in $3!$ ways. Total $= 2 	imes 3! = 12$. For the set ${0, 2, 3, 4}$,the thousands place can be $2, 3,$ or $4$ ($3$ choices). The remaining $3$ positions can be filled in $3!$ ways. Total $= 3 	imes 3! = 18$. Total numbers $= 12 + 18 = 30$.

The number of numbers between $2000$ and $5000$ that can be formed with the digits $0, 1, 2, 3, 4$ (repetition of digits not allowed) and are multiples of $3$ is:

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