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?

- 400
- 320
- 328
- 356
- 408

@ INR

**What are the Constraints?**

1. The digits of the 3-digit number should be distinct.

2. The numbers are even numbers.

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**

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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