SOLUTION: (a) How many words can be formed using letter of PEPSUDENT using each letter at most once? i) If each letter must be used, ii) If some or all the letters may be omitted.

Question 1197425: (a) How many words can be formed using letter of PEPSUDENT using each letter at most once?
i) If each letter must be used,
ii) If some or all the letters may be omitted.
Found 3 solutions by MathLover1, math_helper, ikleyn:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!
i) If each letter must be used

"PEPSUDENT" has letters, P, E, S, U,D, N, and T
there are distinct letters to use, and if each letter must be used ones
that is " Position " or
ways
Here's why that formula works:
Choose the 1st letter any of the ways.
Choose the 2nd letter any of the remaining ways.
Choose the 3rd letter any of the remaining ways.
Choose the 4th letter any of the remaining ways.
Choose the 5th letter any of the remaining ways.
Choose the 6th letter either of the remaining ways.
Choose the 7th letter only the remaining way.
That's

ii) If some or all the letters may be omitted.
That's

or

Answer by math_helper(2461)   (Show Source): You can put this solution on YOUR website!

I think the conventional understanding of using "every letter once" means to include the duplicated letters. For this problem, you would permute all 9 letters but then divide by the arrangements of the duplicates:
9!/(2!*2!) = 362880 / 4 = 90720
OF course, this is also assuming each "word" is just a unique arrangement of letters, and not a dictionary word.
----
EDIT:
After re-reading, it is not clear to me. If each letter must be used "at most once" then why did the problem statement give the duplicate letters? The formulation of the problem statement could be better.
Answer by ikleyn(52779)   (Show Source): You can put this solution on YOUR website!
.

How this problem is worded, it is ambiguous: it allows more than one interpretation,
which is strictly prohibited in Math problems.

It is because the "problem's composer" is not familiar with a standard word setting for such problems.

The standard word setting is to say "if repeating of letters is not allowed".

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