How do I randomly generate five selections from this string?

You can declare a List for storing index of stateCapital. And call Collections.shuffle method to make the indexList randomly.

Then you can loop the indexList to show the questions. This is very simple. You just make the following 2 tiny changes.

1. Add the following code: before looping the questions.

   List<Integer> indexList = new ArrayList<Integer>();
   for(int idx =0; idx <  stateCapital.length; idx++)
   {
       indexList.add(idx);
   }
   Collections.shuffle(indexList);
   

2. Make some change for for-loop.

From

 for (int i = 0; i < stateCapital.length; i++)

to

 for(int i : indexList)

if You just need 5 questions, then you can use following code

 for(int i : indexList.subList(0, 5))

Then all of the questions will displayes randomly and there is no need to change other code.

Completely code is as follows:

import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.Scanner;

public class assn9 {

public static void main(String[] args) {
    String[][] stateCapital = { { "Alabama", "Montgomery" },
            { "Alaska", "Juneau" }, { "Arizona", "Phoenix" },
            { "Arkansas", "Little Rock" }, { "California", "Sacramento" },
            { "Colorado", "Denver" }, { "Connecticut", "Hartford" },
            { "Delaware", "Dover" }, { "Florida", "Tallahassee" },
            { "Georgia", "Atlanta" }, { "Hawaii", "Honolulu" },
            { "Idaho", "Boise" }, { "Illinois", "Springfield" },
            { "Indiana", "Indianapolis" }, { "Iowa", "Des Moines" },
            { "Kansas", "Topeka" }, { "Kentucky", "Frankfort" },
            { "Louisiana", "Baton Rouge" }, { "Maine", "Augusta" },
            { "Maryland", "Annapolis" }, { "Massachusettes", "Boston" },
            { "Michigan", "Lansing" }, { "Minnesota", "Saint Paul" },
            { "Mississippi", "Jackson" }, { "Missouri", "Jefferson City" },
            { "Montana", "Helena" }, { "Nebraska", "Lincoln" },
            { "Nevada", "Carson City" }, { "New Hampshire", "Concord" },
            { "New Jersey", "Trenton" }, { "New York", "Albany" },
            { "New Mexico", "Santa Fe" }, { "North Carolina", "Raleigh" },
            { "North Dakota", "Bismark" }, { "Ohio", "Columbus" },
            { "Oklahoma", "Oklahoma City" }, { "Oregon", "Salem" },
            { "Pennslyvania", "Harrisburg" },
            { "Rhode Island", "Providence" },
            { "South Carolina", "Columbia" }, { "South Dakota", "Pierre" },
            { "Tennessee", "Nashville" }, { "Texas", "Austin" },
            { "Utah", "Salt Lake City" }, { "Vermont", "Montpelier" },
            { "Virginia", "Richmond" }, { "Washington", "Olympia" },
            { "West Virginia", "Charleston" }, { "Wisconsin", "Madison" },
            { "Wyoming", "Cheyenne" } };


    List<Integer> indexList = new ArrayList<Integer>();
    for(int idx =0; idx <  stateCapital.length; idx++)
    {
        indexList.add(idx);
    }
    Collections.shuffle(indexList);

    int correctCount = 0;

    //for (int i = 0; i < indexList.size(); i++) {
    for(int i : indexList){
        System.out.println("What is the capital of " + stateCapital[i][0]
                + "?");
        Scanner input = new Scanner(System.in);
        String capital = input.next();

        if (capital.equalsIgnoreCase(stateCapital[i][1])) {
            correctCount++;
            System.out
                    .println("Your answer is correct, the correct count is "
                            + correctCount);

        } else {

            System.out.println("The correct answer should be "
                    + stateCapital[i][1] + " and the correct count is "
                    + correctCount);
        }
    }

}
}

Random r = new Random();
Scanner s = new Scanner(System.in);

for (int i = 0; i < 5; i++) {
    int random = r.nextInt(50);
    System.out.printf("What is the capital of %s?\n", stateCapital[random][0]);
    String input = s.next();
    if (input.equalsIgnoreCase(stateCapital[random][1])) {
        // Handle correct answer
    } else {
        // Handle incorrect answer
    }
}

I would be lazy and create a function which returns x number of unique random values from within a range:

private static final int[] randomIndices(int datasize. int howmany) {
    if (howmany > datasie) {
        throw new IllegalArumentException();
    }
    int[] indices = new int[howmany];
    for (int i = 0; i < howmany; i++) {
        boolean ok = true;
        do {
            ok = true;
            indices[i] = (int)(datasize * Math.random());
            for (int j = 0; ok && j < i; j++) {
                if (indices[j] == indices[i]) {
                    ok = false;
                }
            }
        } while (!ok);
    }
    return indices;
}

Then, in your main method, I would do:

int[] toask = randomIndices(statecapitals.length, 5);
for (int i = 0; i < toask.length; i++) {
     int question = toask[i];
     ......
     // ask question, etc.
}

To pick 5 random indexes:

List<Integer> indexes = new ArrayList<Integer>(); // create a list to hold all indexes
for (int i = 0; i < stateCapital.length; i++) // populate the list with all indexes
    indexes.add(i);
Collections.shuffle(indexes); // use API to randomly order (shuffle) them
for (Integer i : indexes.subList(0, 5)) { // loop over the first 5
    // your current loop body here
}