A presidential candidate plans to begin her campaign by visiting the capitals in 3 of 41 states. What is the probability that she selects the route of three specific capitals?
$\mathrm{P}($ she selects the route of three specific capitals $)=$
(Type an integer or a simplified fraction.)
Answer
Final Answer: The probability that she selects the route of three specific capitals is \(\boxed{9.380863039399625 \times 10^{-5}}\).
Step 1 ：A presidential candidate plans to begin her campaign by visiting the capitals in 3 of 41 states. We need to find the probability that she selects the route of three specific capitals.
Step 2 ：The probability of selecting a specific route of three capitals out of 41 states is calculated by dividing the number of ways to select the specific route by the total number of ways to select any three capitals.
Step 3 ：The number of ways to select the specific route is 1, because there is only one way to select a specific sequence of three capitals.
Step 4 ：The total number of ways to select any three capitals is calculated by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial. In this case, n is 41 (the total number of states) and k is 3 (the number of states to visit).
Step 5 ：Using the combination formula, we find that the total number of ways to select any three capitals is 10660.
Step 6 ：Finally, we calculate the probability by dividing the number of ways to select the specific route (1) by the total number of ways to select any three capitals (10660). This gives us a probability of \(9.380863039399625 \times 10^{-5}\).
Step 7 ：Final Answer: The probability that she selects the route of three specific capitals is \(\boxed{9.380863039399625 \times 10^{-5}}\).
