Answer:
Explanation:
The Alphabet contains 26 characters, and to better understand this probability, let's do each letter one step at a time.
Any letter being chosen should have an equal probability of:
, and for the first letter we don't really have to worry about choosing a non-repeated letter, since it's impossible to have a non-repeated letter on the first letter... since it's the first letter, so no other letter has been chosen.
This means we have a probability of: 1 to get a non-repeated letter when we choose the first letter.
The probability when we choose the 2nd letter is different, since there is a possibility of choosing a repeated letter. As stated above, the probability of choosing any one letter should be equal for each letter and is:
Since we want to avoid this one letter we choose in the beginning, the probability of choosing it is:
, which means the probability of not choosing it is:
For the 3rd letter it follows the same pattern, assuming we have two non-repeated letters, the probability of choosing a repeated letter from the previous two is there combined probabilities of:
. Since we want a non-repeated letter, we subtract this probability from one, because the entire probability will equal one.
This gives us a probability of:
of choosing a non-repeated letter.
You'll notice a pattern, the numerator is decreasing by one! This makes sense since each time we choose a non-repeated letter, well the number of non-repeated letters will decrease by one.
This gives us the probability for each letter up to eight letters
To get an eight letter code that has no no repeated letters, all these events have to occur. To find the collective probabilities of all these events happening, we simply multiply them, assuming they're independent which we have no reason to believe they aren't independent.
This gives us a total probability of approximately:
