We have 9 letters 3 a's, 2 b's and 4 c's. These 9 letters can be arranged in 9! / (3! 2! 4!)

Correct Option: A

We have 9 letters 3 a's, 2 b's and 4 c's. These 9 letters can be arranged in 9! / (3! 2! 4!) = 1260 ways.

Required number of ways = (2 + 1) (3 + 1) (4 + 1) - 1 = 59

Correct Option: D

Required number of ways = (2 + 1) (3 + 1) (4 + 1) - 1 = 59

12 persons can be seated around a round table in 11! ways. The total number of ways in which 2 particular persons sit side by side = 10! x 2!.

Correct Option: A

12 persons can be seated around a round table in 11! ways. The total number of ways in which 2 particular persons sit side by side = 10! x 2!.

Hence, the required number of arrangements = 11! - 10! x 2! = 9 x (10!)

Required no. of words = ¹²P₄ x ⁴P₃

Correct Option: D

Required no. of words = ¹²P₄ x ⁴P₃
= 12 x 11 x 10 x 9 x 4 x 3 x 2
= 120(100 - 1) x 24
= 288000 - 2880
= 285120

Required no. of triangles = ¹²C₃ - ⁷C₃ = 185

Correct Option: A

Required no. of triangles = ¹²C₃ - ⁷C₃ = 185