# GMAT 650 Level Arithmetic Practice Question 39

#### GMAT Number Properties | Combinatorics | GMAT Sample Questions

The given question is a GMAT hard math problem solving question. This GMAT practice question tests concepts in Number Properties and Combinatorics

Question 39: How many even 3-digit positive integers exist whose digits are distinct?

1. 400
2. 320
3. 328
4. 356
5. 408

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### Explanatory Answer | GMAT Combinatorics

What are the Constraints?
1. The digits of the 3-digit number should be distinct.
2. The numbers are even numbers.

#### Possibility 1: 3-digit numbers an with odd number in hundreds place

Numbers such as 104, 308, 510, 724, and 942 meet this criterion.
The hundreds place can be any of the 5 values viz., 1, 3, 5, 7, or 9.

Because the numbers are even, the unit digit has to be even.
The units place can be any of the 5 values viz., 0, 2, 4, 6, or 8

The tens place should be different from the values that appear in the hundreds and units place.
So, it has 10 - 2 = 8 possibilities.

Number of such 3-digit numbers = 5 × 8 × 5 = 200

#### Possibility 2: 3-digit numbers with even number in hundreds place

Numbers such as 246, 482, 674, and 892 meet this criterion.
The hundreds place can be one of the 4 even numbers other than 0 viz., 2, 4, 6, or 8.

The units place has to be even and should be different from the digit in the hundreds place.
So, 4 possibilities out of the 5 even values are possible.

The tens place should be different from the values that appear in the hundreds and units place.
So, it has 10 - 2 = 8 possibilities.

Number of such 3-digit numbers = 4 × 8 × 4 = 128

Total Number of such 3-digit positive integers = count of step 1 + count of step 2 = 200 + 128 = 328

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