Permutation and Combination

KABRISTAN has 9 letters with vowels A, I, A (3 vowels). Treat the 3 vowels as one unit. We have 7 items to arrange (K,B,R,S,T,N and the vowel unit). These can be arranged in 7! ways. The 3 vowels within their unit can be arranged in 3!/2! ways (since A repeats twice). Total = 7! × (3!/2!) = 5040 × 3 = 15120. Option B is correct. This uses the permutation formula for arranging items with some items grouped together.

Letters in SACHIN in alphabetical order: A, C, H, I, N, S. Words starting with A, C, H, I, N each give 5! = 120 arrangements, totaling 600. For words starting with SA: second letter position in ACHIN gives: SAB... (24), SAC... (24), SAH... (24), SAI... (24), SAN... (24), then SACHIN. Total: 600 + 120 = 601.

Since all 6 letters in BOUNTY are distinct, the number of arrangements is 6 factorial (6!) = 6 × 5 × 4 × 3 × 2 × 1 = 720.

The word JAIHANUMAN has 10 letters with 3 A's and 2 N's repeating. Number of arrangements = 10!/(3! × 2!) = 3628800/(6 × 2) = 3628800/12 = 302400. Option D (302400) is correct.

The word CONSTRUCT has 9 letters with C and T each repeated twice. Total arrangements = 9! / (2! × 2!) = 362880 / 4 = 90720. This formula accounts for duplicate letters that would otherwise create identical arrangements.

The word LUCKNOW has 7 letters with 2 vowels (U, O) and 5 consonants (L, C, K, N, W). Treat the 2 vowels as one unit. Arrangements of 6 units = 6! = 720. Vowels can be arranged within their unit in 2! = 2 ways. Total = 720 × 2 = 1440. Option B is correct.

The word CLERK has 5 distinct letters. Number of arrangements = 5! = 5×4×3×2×1 = 120. Option B is correct.

The word DIMINISH has 8 letters with the letter I repeated 3 times. When arranging letters with repetitions, we divide the total factorial by the factorial of repeated letters: 8!/3! = 40320/6 = 6720. Option D (5040) would be the answer if all letters were unique (8! = 40320 is incorrect).

The word LIGHT has 5 distinct letters. Number of arrangements = 5! = 5*4*3*2*1 = 120. Option E is correct.

The word INHALE has 6 distinct letters. The number of arrangements is 6! = 6×5×4×3×2×1 = 720. Since all letters are different, we don't need to divide by anything.

The word DELIGHT has 7 distinct letters (D, E, L, I, G, H, T). Number of arrangements = 7! = 7×6×5×4×3×2×1 = 5040. Option A is correct.

The word REGION has 6 distinct letters, all different. The number of arrangements is 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Option A (120) is 5!, option C is 7!, and option B and E are incorrect multiplications. This is a fundamental permutation problem.

The word ALGEBRA has 7 letters, but the letter A is repeated twice. The number of distinct arrangements is 7!/2! = 5040/2 = 2520. Option A (5040) is incorrect because it doesn't account for the repetition. Option C (1260) would be 7!/4!, which is not the case here.

The word 'CREATE' has 6 letters, with 'E' appearing twice. Formula: 6! / 2! = 720 / 2 = 360. When arranging letters of a word with repeated letters, divide the factorial of total letters by the factorial of the count of each repeated letter. This accounts for identical arrangements caused by repeated letters.

The word MARKET has 6 distinct letters. All letters are different, so arrangements = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Option B is correct. Option A (1440) would be for 7 items, D (5040) is 7!, C (120) is 5!.