The number of integers between 2,000 and 5,00 | vedclass

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Question

(A) number is divisible by $3$ if the sum of its digits is divisible by $3$.The thousands place can be filled with $2, 3,$ or $4$ because the number m

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(A) number is divisible by $3$ if the sum of its digits is divisible by $3$.The thousands place can be filled with $2, 3,$ or $4$ because the number must be between $2,000$ and $5,000$.Case $1$: Thousands place is $2$.Possible sets of remaining $3$ digits from ${0, 1, 3, 4}$ such that the sum is divisible by $3$:- ${0, 1, 3}$ (sum $= 2+0+1+3 = 6$)- ${0, 3, 4}$ (sum $= 2+0+3+4 = 9$)Each set can be arranged in $3! = 6$ ways. Total $= 2 	imes 6 = 12$.Case $2$: Thousands place is $3$.Possible sets of remaining $3$ digits from ${0, 1, 2, 4}$ such that the sum is divisible by $3$:- ${0, 1, 2}$ (sum $= 3+0+1+2 = 6$)- ${0, 2, 4}$ (sum $= 3+0+2+4 = 9$)Each set can be arranged in $3! = 6$ ways. Total $= 2 	imes 6 = 12$.Case $3$: Thousands place is $4$.Possible sets of remaining $3$ digits from ${0, 1, 2, 3}$ such that the sum is divisible by $3$:- ${0, 2, 1}$ (sum $= 4+0+2+1 = 7$ - No)- ${0, 2, 3}$ (sum $= 4+0+2+3 = 9$)- ${1, 2, 3}$ (sum $= 4+1+2+3 = 10$ - No)Only ${0, 2, 3}$ works. Total $= 1 	imes 6 = 6$.Total numbers $= 12 + 12 + 6 = 30$.

The number of integers between $2,000$ and $5,000$ that can be formed using the digits $0, 1, 2, 3, 4$ (repetition of digits is not allowed) such that the number is a multiple of $3$ is:

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