• Find the probability that if we sample with replacement (ı.e. repetitions are allowed) 6 letters of a 26-letter alphabet, we find a string with 3 different letters each repeated twice (e.g. ”aabbcc”, ”acabcb”, ”xyxyzz”, ”zxyyzx”, etc.)
I can understand the final outcome of this probability but I cannot think of the methodology although I have solved a lot of questions related to combinations, permutations, shuffling,partitions with the multiplication rules and many counting techniques (star and bars, allocation and sampling with and without replacement and with or without order..) However, whenever I face a new question the methodology changes the logic changes too and I have to take into consideration some new details and restrictions.I am still not able to have a big clear image of those techniques in my head and whenever I face a combinatorics problem I am stuck. can you guide me on hon I should think? Maybe it can be less complicated than imagined. ps : I practiced too much and posted many questions on math exchange(but still depending on the solution or a help every time does not solve the problem).
This is so far the table that I know about combinatorics and I can understand the conceptual part of each formula but cannot apply it to the problem I know. I’ll give you below some idea about the kind and level of problems that we tackle in combinatorics. enter image description here
Solving these kinds of problems comes with practicing a lot of them.
When I answer these, I tend to use the word "pick" a lot.
For the question you posed, I'd pick the three repeated letters $_{26}C_3$. For the first letter alphabetically, I'd pick the two slots of the six where they go $_6C_2$. Then I'd pick the two of the four remaining slots where the next letter alphabetically goes $_4C_2$. (The last two letters I have no choice.)
That counts the strings that satisfy what I'm looking for (three different but repeated letters).
For any string, I'd pick the first letter $(26)$, the second letter, $(26)$, and so forth, for a total of $26^6$ possible strings.
Dividing the number of strings I'm looking for by the total number of possible strings gives the answer (B).