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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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