A presidential candidate plans to begin her campaign by visiting the capitals in 4 of 45 states. What is the probability that she selects the route of four specific capitals? P (she selects the route of four specific capitals) (Type an integer or a simplified fraction.)

To find the probability that the presidential candidate selects the route of four specific capitals out of 45 states, we can use the concept of combinations. The total number of ways to select 4 capitals out of 45 states is given by the combination formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where: - \( n \) is the total number of items to choose from (45 states) - \( k \) is the number of items to choose (4 capitals) - \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \) The number of ways to select 4 capitals out of 45 states is: \[ C(45, 4) = \frac{45!}{4!(45-4)!} \] The number of ways to select the route of four specific capitals is 1, as there is only one specific route. Therefore, the probability of selecting the route of four specific capitals is: \[ P(\text{selecting the route of four specific capitals}) = \frac{1}{C(45, 4)} \] Let's calculate this probability. Simplify the expression by following steps: - step0: Solution: \(\frac{1}{\left(\frac{45!}{\left(4!\times \left(45-4\right)!\right)}\right)}\) - step1: Remove the parentheses: \(\frac{1}{\frac{45!}{4!\times \left(45-4\right)!}}\) - step2: Subtract the numbers: \(\frac{1}{\frac{45!}{4!\times 41!}}\) - step3: Reduce the fraction: \(\frac{1}{\frac{45\times 44\times 43\times 42}{4!}}\) - step4: Multiply the terms: \(\frac{1}{\frac{3575880}{4!}}\) - step5: Divide the terms: \(\frac{1}{148995}\) The probability that the presidential candidate selects the route of four specific capitals out of 45 states is approximately \( \frac{1}{148995} \) or about \( 6.711635 \times 10^{-6} \).