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  4. The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is :

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Q. The number of numbers between $2,000$ and $5,000$ that can be formed with the digits $0, 1, 2, 3, 4$ (repetition of digits is not allowed) and are multiple of $3$ is :

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Solution:

Number between 2,000 and 5,000 is 4 digits number.
Using $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of 3 .
For multiple of 3 , the sum of all digits should be divisible by $3 $
So, number can be formed $0,1,2,3$ (sum is 6 which is divisible by 3) or $0,2,3,4$ (sum is 9 which is divisible 1 cannot be on highest digit in the number.
Therefore, number of 4 digit numbers $=2 \times 3 !+3 \times 3 !=30$