1. **State the problem:** Calculate the value of $\binom{400}{2} \times 500$. 2. **Recall the formula for combinations:** $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ This formula gives the number of ways to choose $k$ items from $n$ without regard to order. 3. **Apply the formula for $\binom{400}{2}$:** $$\binom{400}{2} = \frac{400!}{2!(400-2)!} = \frac{400 \times 399}{2 \times 1}$$ We simplify factorials by canceling terms. 4. **Calculate $\binom{400}{2}$:** $$\binom{400}{2} = \frac{400 \times 399}{2} = 200 \times 399 = 79800$$ 5. **Multiply by 500:** $$79800 \times 500 = 39900000$$ 6. **Final answer:** $$\binom{400}{2} \times 500 = 39900000$$ This means there are 39,900,000 ways when combining the selection and multiplication as described.