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Question

The number of 5-digit numbers which are divisible by 4 and sum of digits is odd, with the digits from the set {1,2,3,4,5,6} and repetition of digits is not allowed, is

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Solution

Since the given number is divisible by 4,
therefore, the last two digits of the number must be 12,16,32,32,36,52,56,24 or 64.
Also, the sum of digits of the required number is an odd number. So, three of its digit must be odd and rest two even.

If the number ends in 12,16,32,32,36,52 or 56, then out of remaining three digits 2 will be odd and one will be even.
This can be done in number of ways =2C2×2C1×3!×6=72

If the number ends in 24 or 64, then remaining three digits will be odd.
This can be done in number of ways =3C3×3!×2=12

Total number of numbers will be 84

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