How Many 3-digit Positive Integers With Distinct Digits Are There, Which are not Multiples of 10? GMAT Problem Solving

Question: How many 3-digit positive integers with distinct digits are there, which are not multiples of 10?

1. 576
2. 520
3. 504
4. 432
5. 348

‘How many 3-digit positive integers with distinct digits are there, which are not multiples of 10?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2022”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

It is asked in the question to find out how many 3-digit positive integers having distinct digits are there which are not a multiple of 10
This is a question from permutation and combination.
In order to solve the problem, the candidate should remember the following formulas for permutation and combination.

Permutation of n objects taken r at a time: nPr = n! / (n-r)!
Combination of n objects taken r at a time: nCr = n! / ( (n-r)! * r!)

where x! = x*(x-1)*(x-2)*......(3).(2).(1)
Now we need to find three-digit numbers each of which is going to be distinct
This is the same as the number of ways of filling three spaces with the digits such that no digit is repeated.

The first digit can be filled in 9 ways from numbers 1-9 as 0 cannot be the first digit.
The last digit can be filled in 8 ways from numbers 1-9 as 0 and the first place number cannot be the third digit.
The second digit can be filled in 8 ways from numbers 1-10 except the two numbers already used.
The total number of 3-digit positive integers having distinct digits are there which are not a multiple of 10 = 8*8*9 = 576
Therefore the correct answer is option A.

Correct Answer: A

Approach Solution 2:

It is asked in the question to find out how many 3-digit positive integers having distinct digits are there which are not a multiple of 10.
There are 3 places where digits are to be placed

9C1 options are there fill the first digit
8C1 options are there to fill the second digit
8C1 options are there to fill up the third digit
Total number of ways = 9C1 * 9C1 * 8C1 = 9*8*8 = 576

Therefore the correct answer is option A.

Correct Answer: A

Approach Solution 3:

It is given in the problem that a number not to be a multiple of 10 should not have the units digit of 0.
Let the numbers be:
Case 1: 9 options for the first digit (from 1 to 9 inclusive).
Case 2: 8 options for the third digit (from 1 to 9 inclusive minus the one we used for the first digit).
Case 3: 8 options for the second digit (from 0 to 9 inclusive minus 2 digits we used for the first and the third digits)
Hence, 9*8*8=576.
Correct Answer: A

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