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Question

# How many 4 digit numbers are there, without repetition of digits, which are divisible by 5?

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Solution

## We know that, a number is divisible by 5 if and only if it has 0 or 5 at its units place.Hence, there are two cases:1st case: 4 digit Number ending with 0.–––0–By the method of permutation and combination,The unit place can be filled with only 1 number, that's 0.The thousands place can be filled with remaining 9 numbers.The hundreds place can be filled with remaining 8 numbers.The tens place can be filled with remaining 7 numbers. So, total number of four digit numbers, without repetition, having 0 at their units place are 9×8×7×1=504.2nd case: 4 digit Number ending with 5.–––5–The unit place can be filled with only 1 number. that's 5.The thousands place cannot have 0 and hence, can be filled with remaining 8 numbers. The hundreds place can be filled with remaining 8 numbers. [This can have 0]The tens place can be filled with remaining 7 numbers. So total number of four digit numbers, without repetition, having 5 at their units place are 8×8×7×1=448.Hence, total number of four digit numbers, without repetitions, which are divisible by 5 are 504+448=952.

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