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The question asks for the probability that a presidential candidate selects a specific route by visiting 3 capitals out of 43 available state capitals.

To solve this, let's break down the problem:

1. **Total possible ways to select 3 capitals from 43**: 
   This is a combination problem because the order in which the capitals are chosen doesn't matter. The number of ways to choose 3 capitals from 43 is given by the combination formula:

   \[
   C(n, k) = \frac{n!}{k!(n-k)!}
   \]

   where \( n = 43 \) (total number of capitals) and \( k = 3 \) (capitals chosen). So we need to compute \( C(43, 3) \).

2. **Favorable outcome**: 
   There is only **1 specific route** involving 3 specific capitals.

3. **Probability**: 
   The probability \( P \) is the ratio of the favorable outcomes to the total possible outcomes:

   \[
   P = \frac{1}{C(43, 3)}
   \]

Let me compute this.The total number of ways to select 3 capitals from 43 is 12,341.

Thus, the probability that the presidential candidate selects the route of the 3 specific capitals is:

\[
P = \frac{1}{12,341}
\]

This is the probability in its simplest form.

Would you like further details on this calculation, or do you have any questions?

Here are 5 related questions to deepen understanding:
1. How does the probability change if the number of capitals increases to 50?
2. What would happen if the candidate wanted to visit 4 capitals instead of 3?
3. How does the combination formula differ from the permutation formula, and when should each be used?
4. Can we calculate probabilities if the order of visiting capitals matters?
5. How would the probability change if the candidate can visit the same capital more than once?

**Tip:** In probability, combinations are used when the order doesn't matter, while permutations are used when the order does.

A presidential candidate plans to begin her campaign by visiting the capitals in 3 of 43 states. What is the probability that she selects the route of three specific capitals?

Learn how to calculate the probability that a presidential candidate selects a route of three specific capitals from 43 available state capitals using combinations. This problem covers key concepts in probability and combinations, suitable for students in Grades 10-12.

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