Permutation and Combination

A four-letter word has 4 slots. The first and last slots must be vowels (A, E, I, O, U), so there are 5 choices for each. The middle two slots can be any of the 26 letters of the alphabet. Thus, the total number of words is 5 * 26 * 26 * 5 = 16,900.

Total questions = 6. Must answer 4. Questions 1 and 2 are compulsory, so we must choose 2 more from the remaining 4 questions. Number of ways = 4C2 = (4 * 3) / 2 = 6.

Treat the pair of books as one unit. Now we have 11 units to arrange, which is 11!. Within the pair, the 2 books can be arranged in 2! ways. Total ways = 11! * 2!.

Selecting 6 men from 10 is 10C6. Selecting 5 women from 12 is 12C5. Total ways = 10C6 * 12C5.

Since the lodges are identical, we are partitioning 7 people into groups of 4 and 3. The number of ways is 7C4 = 35. Because the lodges are identical, dividing by 2! is not required here because the group sizes (4 and 3) are distinct, making the partitions inherently unique.

Total letters = 11. M=1, I=4, S=4, P=2. Permutations = 11! / (4! * 4! * 2!) = 39916800 / (24 * 24 * 2) = 39916800 / 1152 = 34650.

MATHEMATICS has 11 letters: M, A, T, H, E, M, A, T, I, C, S. Vowels: A, E, A, I (4). Consonants: M, T, H, M, T, C, S (7). Positions of vowels and consonants are fixed. We need to arrange the 7 consonants (M, T, H, M, T, C, S). Total arrangements = 7! / (2! * 2!) = 5040 / 4 = 1260. The question asks for arrangements of remaining letters, but the calculation 15120 suggests a different interpretation or error.

Total ways to arrange 8 persons = 8!. Ways where the specific person is at an end = 2 * 7!. Total ways - restricted ways = 8! - 2 * 7! = 8 * 7! - 2 * 7! = 6 * 7!.

8 members, 4 males, 4 females. Alternate positions: M F M F M F M F or F M F M F M F M. For each pattern, 4! ways to arrange males and 4! ways for females. Total = 2 * (4! * 4!) = 2 * (24 * 24) = 2 * 576 = 1152.

The word PUNISHMENT has 10 letters: P, U, N, I, S, H, M, E, N, T. There are 7 consonants (P, N, S, H, M, N, T) and 3 vowels (U, I, E). To start and end with a consonant, choose 2 from 7 consonants (7*6 = 42 ways). The remaining 8 letters can be arranged in 8! ways. However, the letter N repeats twice, so divide by 2!. Total = 42 * 8! / 2 = 21 * 40320 = 846720.

Treat the three vowels as one block. The block and the four consonants can be arranged in 5! ways, while the vowels can be arranged within the block in 3! ways, giving 5! x 3!/7! = 1/7.