if a license plate consists of 2 letters followed by 4 digits, how many different plates could be created having at least one letter or digit repeated?
I know its not 6,760,000
Your asking for how many combinations and not the probability so ignore the answer above.
First find the number of combinations if all letters and numbers can be repeated
(26*26*10*10*10*10)=6760000
Then find the number of combinations if none were repeated
(26*25*10*9*8*7)=3276000
(value of combinations that can be repeated)-(value of combinations that no letter or number could be repeated) = 6760000-3276000= 3484000
Well, that's a tricky question! Let's see if we can figure it out with a touch of humor.
First, we'll calculate the total number of possible plates without any restrictions. Since we have 26 letters and 10 digits, we can have 26 * 26 * 10 * 10 * 10 * 10 = 6,760,000 different plates.
Now let's subtract the number of plates where no letter or digit is repeated. We can start by counting the cases where both letters and digits are unique. We have 26 choices for the first letter, 25 for the second (since no repetition is allowed), 10 choices for each digit, and 9 for each subsequent digit (since repetition is not allowed). So, the number of plates with no repetition is 26 * 25 * 10 * 9 * 9 * 9 = 14,130,000.
Finally, we subtract this number from the total number of plates: 6,760,000 - 14,130,000. But wait, this is a negative number! Seems like we made a mistake somewhere.
So, the answer must be less than 6,760,000, but I'm afraid I don't have the exact number for you. However, what I can say for sure is that it's a lot fewer than you expected!
To calculate the number of different license plates that could be created with at least one letter or digit repeated, we can use the principle of inclusion-exclusion.
Step 1: Calculate the total number of possible license plates:
Since the license plate consists of 2 letters followed by 4 digits, there are 26 letters in the English alphabet (excluding the letters I and O) and 10 digits (0-9). This gives us a total of (26 x 26) x (10 x 10 x 10 x 10) = 676,000.
Step 2: Calculate the number of license plates with no repeated letters or digits:
To calculate this, we need to determine the number of ways to choose 2 distinct letters from 26, which is given by the combination formula C(26,2) = 26! / (2!(26-2)!) = 325. Similarly, there are 10 choices for each digit, so the total number of license plates without repeated letters or digits is 325 x 10 x 10 x 10 x 10 = 3,250,000.
Step 3: Calculate the number of license plates with no repeated letters:
To calculate this, we need to determine the number of ways to choose 2 distinct letters from 26, as we did in Step 2. Thus, the number of license plates without repeated letters is 325 x 10 x 10 x 10 x 10 = 3,250,000.
Step 4: Calculate the number of license plates with no repeated digits:
Similar to the previous step, we now need to determine the number of ways to choose 2 distinct digits from 10, which is given by the combination formula C(10,2) = 10! / (2!(10-2)!) = 45. Thus, the number of license plates without repeated digits is 45 x 26 x 26 = 30,120.
Step 5: Apply the inclusion-exclusion principle:
Now, we can calculate the number of license plates that have at least one letter or digit repeated by subtracting the number of license plates without repeated letters (Step 3) and the number of license plates without repeated digits (Step 4) from the total number of possible license plates (Step 1). Thus, the number of license plates with at least one letter or digit repeated is 676,000 - 3,250,000 - 30,120 = 642,880.
Therefore, there are 642,880 different license plates that could be created with at least one letter or digit repeated.
To find the number of different license plates that could be created, we can consider the cases separately for plates that have repeated letters and plates that have repeated digits.
Case 1: Repeated Letters
If a license plate has repeated letters, we need to calculate the number of plates with repeated letters and subtract it from the total number of plates without any restrictions.
To calculate the number of plates without any restrictions:
- The number of choices for the first letter is 26 (26 letters in the alphabet).
- The number of choices for the second letter is also 26.
- The number of choices for each digit is 10 (0-9).
Therefore, the total number of plates without any restrictions is 26*26*10*10*10*10 = 6,760,000.
To calculate the number of plates with repeated letters:
- The number of choices for the repeated letter is 26.
- The number of choices for the other letter is 25 (since we can't repeat the same letter).
Therefore, the number of plates with repeated letters is 26*25*10*10*10*10 = 6,500,000.
Case 2: Repeated Digits
If a license plate has repeated digits, we can use a similar approach as above:
- The number of choices for each letter is 26.
- The number of choices for the first repeated digit is 10.
- The number of choices for the other digits is 9 (since we can't repeat the same digit).
Therefore, the number of plates with repeated digits is 26*26*10*9*9*9 = 5,838,360.
Now, to find the total number of plates with at least one repeated letter or digit, we need to subtract the number of plates without any restrictions from the sum of plates with repeated letters and plates with repeated digits:
6,760,000 - 6,500,000 + 5,838,360 = 6,098,360.
So, there are a total of 6,098,360 different plates that could be created with at least one letter or digit repeated.
consider the Pr of two letters the same: 1/26
Now consider the Pr of two numbers: 1/10*9/10*8/10
Now the Pr of three numbers the same:
1/10*1/10*9/10
now the Pr of four numbers the same:
1/10*1/10*1/10
now the pr of having two pairs of numbers the same:
1/10*9/10*1/9
Pr(at least one letter or digit repeated:
sum of (two letters, no numbers) + (two letters, two numbers)+(two letters, three numbers)+(two letters, four numbers)+(two letters, two pairs of numbers) + (no letters, two numbers)+(no letters, threenumbers)+(no letters, four numbers)+ (no letters,two pairs of numbers)
I will do the first four probabilities for you:
Pr(twoletters, no numbers)=1/26*9/10*8/10*7/10
Pr(two letters, two numbers)=1/26*1/10*9/10*8/10
Pr(two letters, three numbers)=1/26*1/10*1/10*9/10
See if you can finish.