Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is k/2¹⁵, then k is equal to :

Solution

Maths sir · 3 steps

Equal binomial coefficients C(n, 7) = C(n, 9) force 7 + 9 = n, so the die is rolled 16 times.

1. Set up binomial. Odd and even are equally likely on a die, so each roll is a trial with p = q = 1/2. Then P(r odd) = C(n, r)(1/2)ⁿ.
2. Find n. The factor (1/2)ⁿ is the same on both sides, so the condition becomes C(n, 7) = C(n, 9). Since 7 ≠ 9, this needs 7 + 9 = n, so n = 16.

   P(even twice) = C(16, 2)(1/2)¹⁶ = 120/2¹⁶ = 60/2¹⁵

3. So. Matching with k/2¹⁵ gives k = 60, which is option (b).

The idea to keep. When p = q = 1/2, the powers of 1/2 cancel and only the binomial coefficients matter. Use C(n, a) = C(n, b) with a ≠ b, which gives n = a + b.

Answer from the NTA final answer key. Solution by PrepWiser.