Binomial Distribution

The binomial distribution governs the number of successes in a fixed number of independent trials each with the same success probability, giving P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ. This is the canonical JEE Advanced repeated-Bernoulli-trials model. Here n = 5, p = 1/3 for heads, 1 − p = 2/3, and we need exactly r = 2 heads. Thus P(X = 2) = C(5, 2)(1/3)²(2/3)³ = 10 × (1/9) × (8/27) = 10 × 8 / 243 = 80/243. Option 40/243 drops the combinatorial factor by using C(5,2) = 5 instead of 10. Option 10/243 keeps only the binomial coefficient times a single 1/27 and omits the powers of 2/3 correctly. Option 16/81 ignores the coefficient and mismatches the exponents. The result follows directly from the binomial probability formula, with C(5,2) = 10 counting the arrangements of two heads among five tosses. Plausibility check: 80/243 ≈ 0.329 is a valid probability, and since the expected number of heads is np = 5/3 ≈ 1.67, having exactly two heads is plausibly one of the most likely outcomes, consistent with this moderate value.