How Many Three-Digit Numbers are there such that all Three Digits are Different and the First Digit is Not Zero?

Question: How many three-digit numbers are there such that all three digits are different and the first digit is not zero?

1. 504
2.  648
3. 720
4. 729
5. 810

“How many three-digit numbers are there such that all three digits are different and the first digit is not zero?” –  is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Approach 1

It is asked in the question to find out the count of three-digit numbers such that all three digits are different and the first digit is not zero.
Let there be 3 blank spaces _ _ _ .

We have to fill these spaces such that all three digits are different and the first digit is not zero.
The first digit cannot be 0, so there are 9 options for the first digit.
the second digit can be any number (10 possibilities) but cannot be equal to the first digit. Therefore, there are (10-1) = 9 possibilities for the 2nd digit.

The third digit can be any digit (10 possibilities) but cannot be equal to the first two digits. Therefore, there are (10-2) 8 possibilities for the 3rd digit.

So there are 9 * 9 * 8 = 648 possibilities
B is the correct answer.

Approach 2

It is asked in the question to find out the count of three-digit numbers such that all three digits are different and the first digit is not zero.
All three digits are different and the first digit is not zero.
So the first digit can be filled in 9 ways.
And, the second digit can be filled in 9 ways.
And, the third digit can be filled in 8 ways.
Total ways = 9*9*8
= 648

Hence option (B) is correct answer.

Approach 3

From arrangement concept we can have 9 combinations [Hundred's position]* 9 [Ten's position] * 8[Unit's position]

Hence, the total ways as per the arrangement are:
Total ways = 9*9*8
= 648

Correct Answer: B

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