There are 6 distinct letters of the English alphabet and 4 distinct digits. all possible 6 character apha-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)4320b)64800c)8800d)22000e)None of the aboveCorrect answer is option 'B'. Can you explain this answer?

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There are 6 distinct letters of the English alphabet and 4 distinct digits. all possible 6 character apha-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
e)
None of the above

Correct answer is option 'B'. Can you explain this answer?

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There are 6 distinct letters of the English alphabet and 4 distinct di...

To solve this problem, we need to consider the different possibilities for selecting the 4 letters and 2 digits to form the alphanumeric codes. Let's break down the solution into several steps:

Step 1: Selecting 4 letters
Since there are 6 distinct letters in the English alphabet, we need to determine the number of ways to choose 4 of them. This can be calculated using the combination formula:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of items and r is the number of items to be chosen. In this case, n = 6 (number of letters) and r = 4 (number of letters to be chosen).

C(6, 4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15

Therefore, there are 15 ways to select 4 letters.

Step 2: Selecting 2 digits
Similarly, we need to determine the number of ways to choose 2 digits from the available 4 distinct digits. Applying the combination formula again:

C(4, 2) = 4! / (2!(4-2)!) = 4! / (2!2!) = (4 * 3) / (2 * 1) = 6

There are 6 ways to select 2 digits.

Step 3: Combining the letters and digits
Now that we have determined the number of ways to select the letters and digits separately, we can calculate the total number of alphanumeric codes by multiplying these two values together:

Total number of codes = Number of ways to select letters * Number of ways to select digits = 15 * 6 = 90

However, we need to consider that in each code, the characters must be distinct. This means that we cannot have repeated letters or digits within a code. To account for this, we need to calculate the number of ways to arrange the selected letters and digits within a 6-character code.

Step 4: Arranging the characters
Since each code has 6 characters and we have selected 4 letters and 2 digits, we can arrange them in different ways. The total number of arrangements can be calculated using the permutation formula:

P(n) = n!

where n is the number of items to be arranged. In this case, n = 6.

P(6) = 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720

Therefore, there are 720 ways to arrange the 4 letters and 2 digits within a 6-character code.

Step 5: Calculating the maximum number of codes
To calculate the maximum number of codes that can be generated, we need to multiply the total number of codes (90) by the number of arrangements (720):

Maximum number of codes = Total number of codes * Number of arrangements = 90 * 720 = 64,800

Therefore, the correct answer is option B) 64,800.

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There are 6 distinct letters of the English alphabet and 4 distinct di...

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

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