first let me add a disclaimer, my question is not a duplicate of Find the number of 3-letter words that can be made with letters in alphabetical order.
I am learning combinatorics from "Emunerative Counting by George Martin".
I am listing down my solution vs solution given in book, I fail to see how it is a case of combination with repetition. Any help in helping me understand is greatly appreciated.
My Approach
We pick four distinct letters and then for each of them there is only one out of twenty-four which is in alphabetical order, so the solution should be:
$${26 \choose 4}.1$$
Author's solution (copied verbatim)
$${{26 + 4 -1} \choose 4} *1$$
since there is only 1 way to put
4 given letters in alphabetical order.
Please help me understand how is this a case of combination with repetition? Or why is my approach wrong. Thanks
P.S The question given in the book is (clarifying as per Ross's answer)
How many 4-letter words are there with the letters in alphabetical order?
Best Answer
You are picking four distinct letters to form the word. The author is allowing repeated letters, so you do not count AAAA or AABC but the author does. You did not quote the question so it is not clear whether repeated letters are allowed.