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

 To find the number of words that can be formed using the letters of "EXCELLENT," we can use the formula for permutations.

1. If each letter must be used (without repetition), the word is "EXCELLENT," and there are 8 letters in total. Therefore, the number of words is \(8!\) (8 factorial).

   \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320.\]

   So, there are 40,320 words.

2. If some or all the letters may be omitted, then for each letter, we have two choices: either include it in the word or omit it. Since there are 8 letters, each with two choices, the total number of words is \(2^8\).

   \[2^8 = 256.\]

   So, there are 256 words when some or all of the letters may be omitted.