1. **Problem statement:** We are given a binomial distribution $X \sim b(15, 0.4)$ and asked to find $P(X=7)$. 2. **Formula:** The binomial probability formula is $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $n=15$, $p=0.4$, and $k=7$. 3. **Calculate $P(X=7)$:** $$P(X=7) = \binom{15}{7} (0.4)^7 (0.6)^8$$ 4. **Calculate the binomial coefficient:** $$\binom{15}{7} = \frac{15!}{7! \cdot 8!} = 6435$$ 5. **Calculate powers:** $$(0.4)^7 = 0.0016384$$ $$(0.6)^8 = 0.01679616$$ 6. **Multiply all parts:** $$P(X=7) = 6435 \times 0.0016384 \times 0.01679616 = 0.177$$ **Final answer:** $$\boxed{P(X=7) \approx 0.177}$$