Permutation and Combination

The word LEVEL has 5 letters with L appearing twice and E appearing twice. The number of arrangements is 5! / (2! * 2!) = 120 / 4 = 30.

The word LEADING has 7 letters, including 3 vowels (E, A, I) and 4 consonants (L, D, N, G). Treating the vowels as a single unit, we have 5 items to arrange (the vowel block + 4 consonants), which is 5! ways, and the 3 vowels can be arranged among themselves in 3! ways. Total arrangements = 5! * 3! = 120 * 6 = 720.

PROMISE has 7 letters: 3 vowels (O, I, E) and 4 consonants (P, R, M, S). Arrange consonants first: 4! = 24. There are 5 spaces between/around consonants: P_R_M_S. Choose 3 spaces for vowels: 5P3 = 5 * 4 * 3 = 60. Total = 24 * 60 = 1440.

In MATHEMATICS, there are 11 letters: 4 vowels (A, E, A, I) and 7 consonants (M, T, H, M, T, C, S). The vowels occupy 4 fixed positions and consonants occupy 7 fixed positions. The number of ways to arrange vowels is 4! / 2! = 12, and consonants is 7! / (2! * 2!) = 1260. The question implies a specific constraint on relative positions, but the provided answer 3 * 7! is likely based on a specific interpretation of the permutation count.

Total letters = 7. Let n be the number of similar letters. The number of permutations is 7! / n! = 210. 5040 / n! = 210, so n! = 5040 / 210 = 24. Since 4! = 24, n = 4.

The word GIFT has 4 distinct letters. The number of arrangements is 4! = 4 * 3 * 2 * 1 = 24.

Total letters = 6 (a, b, c, d, e, f). Vowels = 2 (a, e), Consonants = 4 (b, c, d, f). Total ways to pick 3 letters = 6P3 = 120. Ways with no vowels = 4P3 = 24. Ways with at least one vowel = 120 - 24 = 96.

Total arrangements of SUCCESS (7 letters: S=3, C=2, U=1, E=1) is 7! / (3! * 2!) = 420. Arrangements where all S are together (treat SSS as one unit) is 5! / 2! = 60. Subtracting these gives 420 - 60 = 360.