A box contains 5 blue and 4 red marbles. In how many ways can 4 marbles be drawn

so as to include 2 red marbles?

To find the number of ways to draw 4 marbles that include 2 red marbles, we need to consider the combinations of marbles that can be drawn.

In this case, we have 5 blue marbles and 4 red marbles, and we want to choose 2 red marbles and 2 additional marbles (which can be either blue or red).

First, let's calculate the number of ways to select 2 red marbles from the 4 available. This can be done using the combination formula:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of objects to choose from and r is the number of objects to choose.

Using this formula, we can calculate C(4, 2):

C(4, 2) = 4! / (2!(4-2)!) = (4 * 3) / (2 * 1) = 6

So, there are 6 ways to select 2 red marbles from the available 4 red marbles.

Now, let's consider the remaining 2 marbles that can be either blue or red. Since we have 5 blue marbles and 4 red marbles, the total number of choices for each of the remaining two marbles is 5 + 4 = 9.

Thus, the total number of ways to draw 4 marbles that include 2 red marbles is given by multiplying the number of ways to select 2 red marbles by the number of choices for the remaining two marbles:

6 * 9 * 9 = 486

Therefore, there are 486 ways to draw 4 marbles from the box that include 2 red marbles.