Number of possible outcomes in a license plate

52,111

Solution 1

That is incorrect.

First, let's find the number of license tags, total, that can be made with $3$ letters and $3$ digits. This is: $$26\cdot26\cdot26\cdot10\cdot10\cdot10 = 26^310^3$$

Second, look at how many tags can be made without repeating any character (letter or number): $$26\cdot25\cdot24\cdot10\cdot9\cdot8$$

Now, if we must repeat at least one character, then number of tags that satisfy are: $$\text{Total} - \text{ThoseThatDon'tRepeat} = 26^310^3-26\cdot25\cdot24\cdot10\cdot9\cdot8$$

Solution 2

The total number of possibilities is $A = 26^3 \times 10^3$. The number of possibilities with no repeats is $B = 26 \times 25 \times 24 \times 10 \times 9 \times 8$. So the number of possibilities with at least one repeat is just $A-B$.

Solution 3

Let us count the three letter strings with at least one letter repeated. There are $26$ where all three letters are the same. To have two letters the same and one different, you have $26$ choices for the double letter, $25$ choices for the single letter, and $3$ choices for the location of the single letter, so there are $26 \cdot 25 \cdot 3 + 26=1976$ total strings with at least one letter repeated. I'll let you count the three digit strings with at least one repeat-call the number $N$. The logic is the same.

Then we would be tempted to say the final answer is $1976\cdot 10^3 + 26^3N$, because if we have a repeated letter we can pair that set with any string of digits. Unfortunately, we have double counted the cases where you have a repeated letter and a repeated digit, so the final answer is $1976\cdot 10^3 + 26^3N- 1976N$ by the inclusion-exclusion principle

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Updated on August 06, 2022

Comments

  • MethodManX
    MethodManX over 1 year

    If a license plate consists of 3 letters followed by 3 digits and having at least one digit or letter repeated .. How many outcomes are there?

    26 * 26 * 10* 10 * 10 .. Is that right?

  • MethodManX
    MethodManX over 10 years
    wait so whats n?
  • Ross Millikan
    Ross Millikan over 10 years
    @MethodManX: I defined it as the number of three digit strings that have at least one repeat. If you follow my logic about the letter strings you should be able to calculate it. I'll be glad to check your result.
  • apnorton
    apnorton over 10 years
    @RossMillikan Hmmm... I thought I had answered in a valid way, but your answer has me second-guessing myself. Does this mean that my answer is incorrect?
  • TonyK
    TonyK over 10 years
    @anorton: Your answer is identical to mine, so I'm sure it's right :-) That doesn't necessarily mean that Ross's answer is wrong, but it's certainly more complicated!
  • MethodManX
    MethodManX over 10 years
    @ anorton I got 6344000 ?
  • apnorton
    apnorton over 10 years
    @MethodManX That's what I got.
  • Ross Millikan
    Ross Millikan over 10 years
    @TonyK: Yes, we have different approaches. I think yours is easier for this problem, but mine is what came to my mind. In some cases this would be the easier approach. Good to have both in your bag of tricks.
  • MethodManX
    MethodManX over 10 years
    @ anorton thanks! Can you help me with my other question please