How Many 4-Digit Integers are there in Which All 4 Digits are GMAT Problem Solving

Question: How many 4-digit integers are there in which all 4 digits are even?

1. 625
2. 600
3. 500
4. 400
5. 256

Solution and Explanation
Approach Solution (1):

Select an even digit for the thousands digit
Since 0 cannot be the first digit of a 4-digit number, this digit can be 2, 4, 6, or 8
So, we can complete stage 1 in 4 ways
Select an even digit for the hundreds digit
This digit can be 0, 2, 4, 6, or 8
We can complete stage 2 in 5 ways
Select an even digit for the tens digit
We can complete stage 3 in 5 ways
Select an even digit for the unit’s digit
We can complete stage 4 in 5 ways
By the Fundamental Counting Principle FCP, we can complete all 4 stages (and thus a 4-digit) in (4) (5) (5) (5) = 500 ways

Correct option: C

Approach Solution (2):

Positive integers: 2, 4, 6, 8, 0
Let the integers of a 4-digit positive number be ABCD

A can take four values (2, 4, 6, 8)
B can take five values (0, 2, 4, 6, 8)
C can take five values (0, 2, 4, 6, 8)
D can take five values (0, 2, 4, 6, 8)

The total is 5 * 5 * 5 * 4
The answer according to this is 500

Correct option: C

Approach Solution (3):

Even digits: 0, 2, 4, 6, 8
The thousands place can be filled in 4 ways
The hundreds place can be filled in 5 ways
The tens place can be filled in 5 ways
The ones place can be filled in 5 ways
Therefore, total number of numbers = 4 *5 * 5 * 5 = 500

Correct option: C

“How many 4-digit integers are there in which all 4 digits are even?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. 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.

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