Permutation and Combination

Total arrangements = 180. If repeated letters are kept together, treat them as a single block. The number of arrangements is reduced significantly based on the number of repeated letters.

Total arrangements of THERAPY (7 letters, 2 vowels E, A) is 7! = 5040. Treat (EA) as one unit, arrangements = 6! * 2! = 1440. Vowels never together = 5040 - 1440 = 3600.

A four-letter palindrome is of the form ABBA. There are 26 choices for A and 26 choices for B. Total = 26 * 26 = 26^2.

PATLIPUTRA has 10 letters: P, A, T, L, I, P, U, T, R, A. Vowels are A, I, U, A (4). Consonants are P, T, L, P, T, R (6). Vowels are at positions 2, 4, 7, 10. Consonants are at 1, 3, 5, 6, 8, 9. Arrange 4 vowels (A, A, I, U) in 4!/2! = 12 ways. Arrange 6 consonants (P, P, T, T, L, R) in 6!/(2!*2!) = 720/4 = 180 ways. Total = 12 * 180 = 2160.

Total arrangements of 10 people is 10!. In any arrangement, the relative order of A, B, and C can be in 3! = 6 ways. Only one of these ways (A before B before C) is valid. Thus, the number of valid ways is 10! / 3! = 10 * 9 * 8 * 7! / 6 = 10 * 9 * 8 * 7! / 6 = 120 * 7! = 10! / 6.

MATHEMATICS has 11 letters: M, A, T, H, E, I, C, S (8 unique). We need to choose 4 different letters from 8: 8C4 = 70. Each set of 4 can be arranged in 4! = 24 ways. Total = 70 * 24 = 1680.

Total arrangements of 6 papers = 6! = 720. Treat the 2 Math papers as one unit. Arrangements with Math together = 5! * 2! = 120 * 2 = 240. Arrangements with Math not together = 720 - 240 = 480.

Choose 3 consonants from 8: 8C3 = 56. Choose 3 vowels from 5: 5C3 = 10. Total combinations = 56 * 10 = 560. Each combination has 6 letters, which can be arranged in 6! = 720 ways. Total words = 560 * 720 = 403,200.

The word LAUGH has 5 letters: L, A, U, G, H. Vowels are A, U. Treat (AU) as one unit. We have 4 units: {L, G, H, (AU)}. These can be arranged in 4! = 24 ways. The vowels (AU) can be arranged in 2! = 2 ways. Total = 24 * 2 = 48.

Arrange the 3 subjects as blocks: 3! ways. Within each block, books can be arranged: 5! for Physics, 3! for Chemistry, 3! for Maths. Total = 3! * 5! * 3! * 3!.

Total 10 people (6 teachers, 4 boys). Committee of 8. At least 4 teachers: (4T, 4B) or (5T, 3B) or (6T, 2B). Ways = C(6,4)*C(4,4) + C(6,5)*C(4,3) + C(6,6)*C(4,2) = 15*1 + 6*4 + 1*6 = 15 + 24 + 6 = 45.

Arranging 3 red and 3 blue berries in a row alternately: RBRBRB or BRBRBR. Since berries are distinguishable, there are 3! ways to arrange red and 3! ways to arrange blue. Total = 2 * (3! * 3!) = 2 * 6 * 6 = 72.