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.