1. **State the problem:** We want to find the probability of exactly $x=1$ successes in $n=100$ trials with success probability $p=0.1$ using the binomial distribution. 2. **Formula:** The binomial probability formula is $$P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$$ where $\binom{n}{x} = \frac{n!}{x!(n-x)!}$ is the binomial coefficient. 3. **Calculate the binomial coefficient:** $$\binom{100}{1} = \frac{100!}{1! \times 99!} = 100$$ 4. **Calculate the probability:** $$P(X=1) = 100 \times (0.1)^1 \times (0.9)^{99}$$ 5. **Evaluate powers:** $$0.1^1 = 0.1$$ $$0.9^{99} \approx 0.00002656$$ 6. **Multiply all parts:** $$P(X=1) = 100 \times 0.1 \times 0.00002656 = 0.0002656$$ 7. **Interpretation:** The probability of exactly one success in 100 trials with success probability 0.1 is approximately 0.00027.