1. **Problem statement:** Calculate the binomial coefficient $\binom{34}{15}$ and explain its meaning in the context of tree diagrams and binomial distribution. 2. **Formula:** The binomial coefficient is given by $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ denotes the factorial of $n$. 3. **Calculation:** $$\binom{34}{15} = \frac{34!}{15! \times (34-15)!} = \frac{34!}{15! \times 19!}$$ 4. **Intermediate simplification:** $$\binom{34}{15} = \frac{34 \times 33 \times \cdots \times 20}{15 \times 14 \times \cdots \times 1}$$ 5. **Meaning:** - In a **tree diagram**, $\binom{34}{15}$ represents the number of ways to choose 15 successes (or specific outcomes) out of 34 trials. - In the **binomial distribution**, it is the coefficient that counts the number of ways to have exactly 15 successes in 34 independent Bernoulli trials. 6. **Final answer:** $$\binom{34}{15} = 818809200$$ This means there are 818,809,200 ways to choose 15 items from 34.