If a fair 6-sided die is rolled 10 times, what is the probability that the face showing '2'...

The binomial probability formula for exactly k successes in n trials is P(X = k) = C(n, k) × p^k × (1−p)^(n−k). Here, n = 10, k = 4, p = 1/6 (probability of rolling a 2), and (1−p) = 5/6. This gives C(10, 4) × (1/6)^4 × (5/6)^6. Option A correctly applies all parameters; B uses k = 2 instead of 4, C swaps the exponents, and D uses k = 5.

Step-by-step Derivation:
Step 1: Identify the binomial distribution parameters.

• n (number of trials) = 10
• k (number of successes) = 4
• p (probability of success on each trial) = 1/6
• q (probability of failure on each trial) = 5/6

Step 2: Apply the binomial probability formula.
P(X = 4) = C(n, k) × p^k × q^(n-k)
P(X = 4) = C(10, 4) × (1/6)^4 × (5/6)^(10-4)
P(X = 4) = C(10, 4) × (1/6)^4 × (5/6)^6

Step 3: Verify the exponents sum correctly.
Exponent sum: 4 + 6 = 10 ✓

Step 4: Eliminate incorrect options.

• Option B: Uses C(10, 2) and (1/6)^2 — wrong k value.
• Option C: Uses C(10, 4) but reverses exponents to (1/6)^6 × (5/6)^4 — exponents don't match n and k.
• Option D: Uses C(10, 5) and k = 5 — wrong k value.

Answer: A