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Question

How many -digit numbers are there with no digit repeated?

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Solution

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A four digit number is to be formed from the digits

Since, leftmost place i.e thousand's place cannot have zero. So, there are ways to fill thousand's place.
Since, repetition is not allowed , so hundreds place can be filled by remaining digits .
So, hundred's place can be filled in ways.
Similarly, to fill tens place, we have digits remaining.
So,tens place can be filled by ways.
Similarly, to fill ones place, we have digits remaining.
So,ones place can be filled by ways.
So, required number of ways in which four digit numbers can be formed from the given digits is

Alternative Method:
The thousands place of the -digit number is to be filled with any of the digits from to as the digit cannot be included. Therefore, the number of ways in which thousands place can be filled is
The hundreds, tens, and units place can be filled by any of the digits from to .
However, the digits cannot be repeated In the -digit numbers and thousands place is already occupied with a digit. The hundreds, tens, and units place is to be filled by the remaining digits. 
Therefore, there will be as many such -digit numbers as there are permutations of different digits taken at a time.

Number of such -digit numbers
 
Thus, by multiplication principle, the required number of -digit numbers is

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