Which expression equals P(X = 3) for X ~ Binomial(n = 10, p = 0.2)?

• A. 120*(0.2)^3*(0.8)^7
• B. 45*(0.2)^3*(0.8)^7
• C. 210*(0.2)^3*(0.8)^7
• D. 10*(0.2)^3*(0.8)^7

Answer: (A)

This uses the binomial probability mass function for exactly k successes in n independent trials with success probability p. P(X=k) = C(n,k) p^k (1−p)^{n−k}. With n=10, p=0.2, and k=3, you get P(X=3) = C(10,3) (0.2)^3 (0.8)^7. Since C(10,3) = 120, that becomes 120*(0.2)^3*(0.8)^7, which matches the expression shown. The other coefficients correspond to different k values (like 10 choose 2 = 45 for k=2, 10 choose 4 = 210 for k=4, and 10 for k=1), which is why they don’t describe exactly three successes.