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A monkey types a 280-character message on Twitter, using only the 26 upper-case English letters. A research assistant is trying to determine if there is any hidden meaning in the message, and tries partitioning the message into 3 segments (the first $x$ letters, the next $y$ letters, and the last $280-x-y$ letters). How many ways could the message be partitioned, assuming each segment has at least one letter?

Attempt at solution:

I was wondering partition 1 would have one character - 1 character, partition 2 would have 1 letter (-1 character) and the third parse would have two letters because they would have x and y (-2 characters). Would it be then 276 distingusible objects (the characters) over 3 distinguishable boxes (the parsese)?

Blue
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Math Whiz
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  • If I understand correctly, you don't care that there are 26 capital letters available, as you aren't asking how many possible messages there are. You are instead asking how many ways there are to split up a $280$ character message into three chunks, or equivalently asking how many solutions in (either positive or nonnegative, it's not clear which) integers there are to $x + y + z = 280$. Is that right? – davidlowryduda Apr 07 '21 at 23:27
  • Yeah I don't think the capital letters matter at this point. – Math Whiz Apr 07 '21 at 23:28
  • It would be 0 < x < y < z = 280? And count the number of solutions to the equation? – Math Whiz Apr 07 '21 at 23:29
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    Commenting rather than answering because I'm not supposed to repeat answers that are prevalent on this website. Anyway, this is a Stars and Bars problem where $x + y + z = 280$, if I understand the query correctly. See Link1 and Link2. Note that if, instead, you are interested in $x + y + z < 280$, then the Hockey Stick Identity is also relevant. – user2661923 Apr 07 '21 at 23:37
  • How many ways can two markers within the line of letters be put to partition them into three segments ? – true blue anil Apr 08 '21 at 04:34

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