An unbiased die is rolled n times. Let P A , P B and P C be the probability of occurrence of an odd number exactly one, two and three times respectively in n trials. If P A , P B , P C are in arithmetic progression, then n is equal to

Step-by-step solution

P A = C n 1 1 2 1 1 2 n - 1 P B = C n 2 1 2 2 1 2 n - 2 P C = C n 3 1 2 3 1 2 n - 3 ⇒ P A , P B , P C are in AP ⇒ 2 × C n 2 = C n 1 + C n 3 ⇒ 2 × n n - 1 2 = n + n n - 1 n - 2 6 ⇒ 2 × n n - 1 2 = n + n n - 1 n - 2 6 ⇒ n - 1 = 1 + n - 1 n - 2 6 ⇒ 6 n - 6 = 6 + n 2 - 3 n + 2 ⇒ n 2 - 9 n + 14 = 0 ⇒ n = 2 or 7 ∴ n = 2 is not possible ⇒ n = 7