Permutation and Combination

Total letters = 3+2+4 = 9. Arrangements = 9! / (3! * 2! * 4!) = 362880 / (6 * 2 * 24) = 362880 / 288 = 1260.

Total letters = 7 (C, L, I, F, T, O, N). Vowels are I, O. Treating (IO) as one unit, we have 6 units to arrange in 6! ways. The vowels can be arranged in 2! ways. Total arrangements = 7!. Probability = (6! * 2!) / 7! = 2/7.

The word 'series' has 6 letters: s, e, r, i, e, s. The distinct letters are s, e, r, i. Since we need to pick 3 letters, we can have cases with repeated letters (s, s, e, e) or all distinct. Calculating permutations for these combinations yields 42.

Word 'FIGMENT' has 7 letters: F, I, G, M, E, N, T. Vowels are I, E. If vowels are at first and last, there are 2! ways to arrange vowels. The remaining 5 letters can be arranged in 5! = 120 ways. Total = 2 * 120 = 240.

Word 'FORTUNATES' has 10 letters: F, O, R, T, U, N, A, T, E, S. Two T's. Total arrangements = 10!/2!. Arrangements with T's together = treat TT as one unit, so 9! arrangements. Probability = 9! / (10!/2!) = 2 * 9! / 10! = 2/10 = 1/5.

MISTAKE has 7 letters: 3 vowels (I, A, E) and 4 consonants (M, S, T, K). To keep vowels apart, place consonants first: _ C _ C _ C _ C _. There are 5 spaces for 3 vowels. Ways = 4! * 5P3 = 24 * 60 = 1440.