How many words can be formed using letter of DEPARTMENT using each letter at most once?

i) If each letter must be used,

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

i) If each letter must be used:

The word "DEPARTMENT" has 10 letters in total, but there are repeating letters. Specifically, the letter "T" appears 2 times, and "E" appears 2 times.

To calculate the number of distinct permutations of the letters, we use the formula for permutations of multiset elements:

Number of distinct permutations=10!2!×2!\text{Number of distinct permutations} = \frac{10!}{2! \times 2!}

Where:

• 10!10! is the total number of ways to arrange the 10 letters,

• 2!2! accounts for the repeated "T"s, _________ _______ _____ ____ _____ ________ ________ ___.
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