The number of numbers between 2,000 and 5,000 | vedclass

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Question

(A) number is divisible by $3$ if the sum of its digits is divisible by $3$. We need to form $4$-digit numbers between $2,000$ and $5,000$ using digit

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(A) number is divisible by $3$ if the sum of its digits is divisible by $3$. We need to form $4$-digit numbers between $2,000$ and $5,000$ using digits ${0, 1, 2, 3, 4}$ without repetition.Case $1$: Thousands place is $2$.Possible sets of remaining $3$ digits such that the sum is a multiple of $3$: ${0, 1, 3}$ (sum $6$),${0, 3, 4}$ (sum $9$),${1, 3, 4}$ (sum $10$ - No),${0, 1, 4}$ (sum $7$ - No).For ${0, 1, 3}$,arrangements $= 3! = 6$.For ${0, 3, 4}$,arrangements $= 3! = 6$.Total for Case $1 = 6 + 6 = 12$.Case $2$: Thousands place is $3$.Possible sets of remaining $3$ digits: ${0, 1, 2}$ (sum $6$),${0, 2, 4}$ (sum $9$),${1, 2, 4}$ (sum $10$ - No).For ${0, 1, 2}$,arrangements $= 3! = 6$.For ${0, 2, 4}$,arrangements $= 3! = 6$.Total for Case $2 = 6 + 6 = 12$.Case $3$: Thousands place is $4$.Possible sets of remaining $3$ digits: ${0, 2, 3}$ (sum $9$),${1, 2, 3}$ (sum $10$ - No).For ${0, 2, 3}$,arrangements $= 3! = 6$.Total for Case $3 = 6$.Total numbers $= 12 + 12 + 6 = 30$.

The number of numbers between $2,000$ and $5,000$ that can be formed using the digits $0, 1, 2, 3, 4$ (repetition of digits is not allowed) which are multiples of $3$ is?

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