Permutation and Combination

The word SOFTWARE has 8 letters: 5 consonants (S, F, T, W, R) and 3 vowels (O, A, E). Treating the 3 vowels as one unit gives us 6 objects to arrange in 6! = 720 ways. The vowels themselves can be arranged in 3! = 6 ways. Total arrangements = 720 × 6 = 4320. None of the numerical options A-D (13440, 1440, 360, 120) are correct, making E ('None of these') the right answer.

THERAPY has 7 distinct letters. Total arrangements = 7! = 5040. When A and Y must be together, treat them as one unit: 6! × 2 = 1440 (arrangements × 2 orders for AY/YA). Therefore, arrangements where A and Y are NOT together = 5040 - 1440 = 3600. Option C is correct.

ARISE has 5 distinct letters. Number of arrangements = 5! = 5 × 4 × 3 × 2 × 1 = 120.

The word SHOULDER has 8 distinct letters. The number of arrangements = 8! = 8×7×6×5×4×3×2×1 = 40320. All letters are unique, so we simply calculate the factorial.

Number of ways to choose 4 books from 11 = C(11,4) = 11!/(4!7!) = (11×10×9×8)/(4×3×2×1) = 7920/24 = 330. Option D is correct. This is a standard combination problem - order doesn't matter.

The word ALIGHT has 6 distinct letters. Number of arrangements = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Since all letters are different, no division for repetitions is needed.

The word PAINT has 5 distinct letters. The number of arrangements is 5! = 5 × 4 × 3 × 2 × 1 = 120. This is the formula for permutations of n distinct objects taken all at once: n!.

The word PERSONAL has 8 letters. Consonants are P, R, S, N, L (5 consonants). Vowels are E, O, A (3 vowels). The last position must be a consonant, so choose 1 of 5 consonants for that position. The remaining 7 positions can be filled with the remaining 7 letters in 7! = 5040 ways. Total arrangements = 5 × 7! = 5 × 5040 = 25200.

The student must attempt 6 questions with at least 2 from each part. The possible selections are: 2 from Part I and 4 from Part II (C(5,2)×C(5,4) = 10×5 = 50), 3 from each (C(5,3)×C(5,3) = 10×10 = 100), or 4 from Part I and 2 from Part II (C(5,4)×C(5,2) = 5×10 = 50). Total = 50+100+50 = 200 ways.

LUCKNOW has 7 letters with 2 vowels (U, O). Treat vowels as one unit: we have 6 units (5 consonants + 1 vowel group). These can be arranged in 6! ways. Within the vowel group, U and O can be arranged in 2! ways. Total arrangements = 6! × 2!.

The word 'STORM' has 5 distinct letters. The number of arrangements is 5! = 5 × 4 × 3 × 2 × 1 = 120. This is a permutation of n distinct objects taken all at a time, calculated as n factorial.

EDUCATION has 9 distinct letters: 4 consonants (D,C,T,N) and 5 vowels (E,U,A,I,O). First position is fixed as 'A' (1 choice). Last position must be a consonant (4 choices). Remaining 7 positions can be filled by remaining 7 letters in 7! ways. Total = 1×4×7! = 4×5040 = 20160. Option A (22140) incorrectly uses 8! instead of 7!, while option C (22160) is a miscalculation.

Total arrangements of 12 papers is 12!. If best and worst are together, treat them as one unit (11! ways to arrange) × 2! ways to order the two within the unit. Subtract from total: 12! - 11!×2! = 11!(12-2) = 10×11!.

TROUBLE has 7 letters with vowels O, U, E (3 vowels) and consonants T, R, B, L (4 consonants). Treat vowels as one block: [OUE] T R B L. This gives 5! = 120 arrangements of blocks. Within the vowel block, 3 vowels can be arranged in 3! = 6 ways. Total = 120 × 6 = 720.

Treat the 4 consonants (D, C, T, N) as one block. Arrange 6 items (5 vowels + 1 consonant block) in 6! = 720 ways. Within the consonant block, arrange the 4 consonants in 4! = 24 ways. Total = 720 × 24 = 17280. Option C is correct.