How Many 5 Digit (digit = 0 - 9) Zip Codes can Exist in Which at Least One Digit is Repeated ?

Question: How many 5 digit (digit = 0 - 9) zip codes can exist in which at least one digit is repeated ?

1. 100,000
2. 90,000
3. 69,760
4. 62,784
5. 30,240

“How many 5 digit (digit = 0 - 9) zip codes can exist in which at least one digit is repeated ?”- 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 number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation: 

Approach Solution 1:

Trick:
A 5 digit ZIP code is entirely different from a 5 digit number. 5 digit number cannot start with 0, as it becomes 4 digit in this case. However, a 5 digit ZIP code can be 01234. Meaning that if we take zip code of a place, it can start with 0 eg 01234 and still be 5 digit.

In the above question, we need to find out:

The number of 5 digit (digit = 0 - 9) zip codes that can exist in which at least one digit is repeated.

Approach:
We will have to check all the possible zip codes with a 5 digit zip. Once we find that, we need to find out zip codes with no digits being repeated.
Once these two are out, we can just subtract them and find the answer.

All possibilities - No digit is repeated

= (10 * 10 * 10 * 10 * 10) - (6 *7 * 8 * 9 * 10)

= 100000 - 42 * 720

= 100000 - 30240

= 69760

Hence, C is the correct answer.

Approach Solution 2:

In the above question, we need to find out:

The number of 5 digit (digit = 0 - 9) zip codes that can exist in which at least one digit is repeated.

Approach:

We will have to check all the possible zip codes with a 5 digit zip. Once we find that, we need to find out zip codes with no digits being repeated.

Once these two are out, we can just subtract them and find the answer.

Zip codes with 5 digits (all five digits can be repeated): 10^5=100,000

Zip codes with no digit being repeated: 10*9*8*7*6=30,240

Zip codes with at least one digit being repeated: 100,000-30,240=69,760

Hence, C is the correct answer.

Approach Solution 3:

The key to solving the equation is thinking other way around and getting the answer by the following method:

We will check the total combinations first:

The total no of combination of both repeated and non repeated terms = (10)^5 = 100000
The total no of combination of non repeated terms= 10*9*8*7*6=30240

Now we will substract:

The total no of combination of repeated terms with at least one digit = 100000-30240=69760

Hence, C is the correct answer. 

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