SOLUTION: A Discrete Mathematics student comes from the village of KILLYIKYAKYODLE. As an hilarious jape, on the way home from a late-night study session, she decides to rearrange the letter

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Question 1001499: A Discrete Mathematics student comes from the village of KILLYIKYAKYODLE. As an hilarious jape, on the way home from a late-night study session, she decides to rearrange the letters on one of the sign-posts.
How many arrangements of the letters in KILLYIKYAKYODLE are possible?
i got 15!/3!*2!*3!*2! for this
Of these arrangements, how many have all the L’s together?
Of these arrangements, how may have all the letters in alphabetical order?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
How many arrangements of the letters in KILLYIKYAKYODLE are possible?
There are
3 instances of the letter K ,
3 instances of the letter Y ,
3 instances of the letter L ,
2 instances of the letter I ,
and 1 each of the 4%7D%7D%7C+letters+%7B%7B%7BA , D , E , and O ,
for a total of 3%2B3%2B3%2B2%2B4=15 characters.
If we could distinguish between the repeated instances of the same letter,
we would see 15%21 different arrangements.
The 3 instances of the letter K can be arranged 3%21=6 different ways.
The same goes for
the 3 instances of the letter Y ,
3 instances of the letter L , and
the 3 instances of the letter Y ,
3 instances of the letter L ,
while the 2 instances of the letter I can be arranged 2%21=2 ways.
So, if the repeats of the same letter cannot be distinguished, we would see
highlight%2815%21%2F%283%21%2A3%21%2A3%21%2A2%21%29%29 arrangements,
which I believe is what you meant to write.

Of these arrangements, how many have all the L’s together?
If we do not include the 3 instances of the letter L ,
we have 15-3=12 characters, that can be arranged into
12%21%2F%283%21%2A3%21%2A2%21%29 12-letter "words."
In each 12-letter "word", we can then intercalate the 3 L’s together in
12%2B1=13 different positions:
before any of the 12 letter positions, or at the end.
We end up with highlight%2813%2A12%21%2F%283%21%2A3%21%2A2%21%29%29 arrangements that have all the L’s together.

Of all the possible arrangements, how may have all the letters in alphabetical order?
To have all the letters in alphabetical order,
the first character must be A,
the second character must be D,
the third character must be E,
the 2 I's must follow as the 4th and 5th characters,
followed by the 3 K's,
then the 3 L's,
followed by the O,
and ending with the 3 Y's.
That is the prescribed order:
highlight%281%29 prescribed arrangement, no choices,
unless we can distinguish between the repeated instances of the same letter.