Question: There are 6 distinct letters of the English alphabet and 4 distinct digits. All possible 6 character alpha-numero codes are generated using any 4 letters of the alphabet and any 2 available digits. If in any given code, the characters are all distinct, then what is the maximum number of such codes that can be generated?

         A. 4320
         B. 64800
         C. 8800
         D. 22000

Answer: B

Approach Solution (1):

4 letters from 6 can be selected in 6C4 ways or 15 ways
2 numbers from 4 can be selected in 4C2 ways or 6 ways
Now, we have selected 6 characters in 15 * 6 = 90 ways
The 6 characters can be arranged in 6! Ways or 720 ways
Therefore, the total number of codes is 90* 720 = 64800
Correct option: B

Approach Solution (2):
Total number of digits in the code is 6
Out of this, 4 are alphabets = 6C4 ways = 15
The other 2 are numeric digits = 4C2 ways = 6
Now these 6 digits can be arranged among themselves in 6! Ways
So the answer is 15*6*6! = 64800
Correct option: B

Approach Solution (3):
Let’s first find the number of codes in the format LLLLDD where L denotes a letter and D denotes a digit. We must choose 4 of 6 letters, and repeats are not allowed for any individual code. Since order matters, there are 6P4 ways to choose the letters. Similarly, from 4 digits, we must choose 2, and repeats in any individual code are not allowed. Since order matters, this can be done on 4P2 ways
Thus, the number of possible digit/code combinations of the format LLLLDD is:
6P4 * 4P2 = (6*5*4*3) * (4*3) = 360*12 = 4,320
Next, let’s find the number of different formats (such as LLDLLD or DDLLLL etc.) that one can create a code. We notice that LLLLDD can be arranged in 6!/(4!*2!) = (6*5)/2 =15 ways
Any one of these 15 formats has the same number of codes are LLLLDD; therefore these are a total of 4,320*15 = 64,800 possible codes.
Correct option: B

“There are 6 distinct letters of the English alphabet and 4 distinct digits. All possible 6 character alpha-numero codes are generated using any 4 letters of the alphabet and any 2 available digits. If in any given code, the characters are all distinct, then what is the maximum number of such codes that can be generated?”- 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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