If license numbers consist of 3 letters followed by 3 digits, how many different licenses could be created having at least one letter or digit repeated?
Well, let's calculate it step by step, shall we?
First, let's calculate the number of licenses that have all unique characters. For the letters, we have 26 options for the first letter, 25 options for the second letter (since it shouldn't be the same as the first), and 24 options for the third letter. For the digits, we have 10 options for each digit (0-9). So, the number of unique licenses is 26 * 25 * 24 * 10 * 9 * 8 = 11,232,000.
Now, let's calculate the number of licenses that have at least one repeated character. To get the total number of licenses, we subtract the number of unique licenses from the total number of possible licenses: (26^3 * 10^3) - 11,232,000 = 16,753,000.
Therefore, there could be a total of 16,753,000 different licenses with at least one letter or digit repeated. That's quite a lot of possibilities!
To calculate the number of licenses that could be created with at least one letter or digit repeated, we can break down the problem into two cases:
Case 1: At least one letter is repeated:
In this case, we need to consider the number of licenses where two letters are repeated, and the number of licenses where three letters are repeated.
Number of licenses with two letters repeated:
To calculate this, we first choose 2 letters out of 26 (since there are 26 letters in the alphabet), and assign them to two of the three letter positions. The remaining position can be filled with any of the 10 digits from 0 to 9. So, the number of licenses with two letters repeated is:
26C2 * 3C2 * 10 = (26 * 25 / 2) * 3 * 10 = 19,500 licenses
Number of licenses with three letters repeated:
In this scenario, we simply need to choose 1 letter out of 26 and assign it to all three letter positions. The remaining positions can be filled with any of the 10 digits. So, the number of licenses with three letters repeated is:
26C1 * 3 * 10 = 780 licenses
Case 2: At least one digit is repeated:
In this case, we need to consider the number of licenses where two digits are repeated, and the number of licenses where three digits are repeated.
Number of licenses with two digits repeated:
Similar to the first case, we select 2 digits out of 10 and assign them to two of the three digit positions. The remaining position can be filled with any of the 26 letters. So, the number of licenses with two digits repeated is:
10C2 * 3C2 * 26 = (10 * 9 / 2) * 3 * 26 = 7,800 licenses
Number of licenses with three digits repeated:
Again, we simply need to choose 1 digit out of 10 and assign it to all three digit positions. The remaining positions can be filled with any of the 26 letters. So, the number of licenses with three digits repeated is:
10C1 * 3 * 26 = 780 licenses
Total number of licenses with at least one letter or digit repeated:
To get the total number of licenses with at least one letter or digit repeated, we add up the numbers from both cases:
19,500 + 780 + 7,800 + 780 = 29,860 licenses
Therefore, there could be 29,860 different licenses created with at least one letter or digit repeated.
To find the number of different licenses that could be created with at least one letter or digit repeated, we can approach this problem using the principle of inclusion-exclusion.
Step 1: Calculate the total number of possible licenses without any restrictions.
Since there are 26 letters in the alphabet and 10 digits (0-9), we have 26 choices for the first letter, 26 choices for the second letter, 26 choices for the third letter, 10 choices for the first digit, 10 choices for the second digit, and 10 choices for the third digit. Multiply these choices together to get the total possibilities:
26 * 26 * 26 * 10 * 10 * 10 = 17,576,000
So, without any restrictions, there are 17,576,000 different licenses that could be created.
Step 2: Calculate the number of licenses that have no repeated letters or digits.
To calculate this, we need to choose 3 distinct letters from the 26 available, and 3 distinct digits from the 10 available. Use combination notation (nCr) to calculate this:
C(26, 3) * C(10, 3) = 26! / (3! * (26-3)!) * 10! / (3! * (10-3)!) = 26 * 25 * 24 * 10 * 9 * 8 = 8,748,000
So, there are 8,748,000 different licenses that have no repeated letters or digits.
Step 3: Calculate the number of licenses that have no repeated letters but may have repeated digits.
To calculate this, we subtract the number of licenses that have no repeated letters or digits from the total number of possibilities. So,
17,576,000 - 8,748,000 = 8,828,000
Step 4: Calculate the number of licenses that have no repeated digits but may have repeated letters.
To calculate this, we subtract the number of licenses that have no repeated letters or digits from the total number of possibilities. So,
17,576,000 - 8,748,000 = 8,828,000
Step 5: Calculate the number of licenses that have repeated letters and digits.
To find this, we need to calculate the number of licenses that have no repeated letters and no repeated digits and subtract it from the total number of possibilities.
For licenses that have no repeated letters and no repeated digits, we have:
C(26, 3) * C(10, 3) = 8,748,000
So, to find the licenses that have repeated letters and digits, we subtract this value from the total possibilities:
17,576,000 - 8,748,000 = 8,828,000
Step 6: Calculate the final result.
Now that we have the number of licenses falling into each category, we can use the principle of inclusion-exclusion to find the final number of licenses that have at least one letter or digit repeated:
Total licenses with repeated letters or digits = (License without repeated letters) + (License without repeated digits) - (License without repeated letters and digits) + (License with repeated letters and digits)
Total licenses with repeated letters or digits = 8,828,000 + 8,828,000 - 8,748,000 + 8,828,000
Total licenses with repeated letters or digits = 17,736,000
Therefore, there could be a total of 17,736,000 different licenses created that have at least one letter or digit repeated.
Number of all possible plates
N1=26³*10³
Number of plates with unique letters AND unique digits
N2=26*25*24*10*9*8
The difference of N1 and N2 gives the number of plates with at least one repeated digit or letter.