A bag contains 2 red marbles, 4 green ones, 1 lavender one, 2 yellows, and 6 orange marbles. How many sets of four marbles include all the red ones?
To find the number of sets of four marbles that include all the red ones, we need to consider the total number of marbles available and the number of ways we can choose four marbles from them.
First, let's count the total number of marbles: 2 red + 4 green + 1 lavender + 2 yellow + 6 orange = 15 marbles.
Next, we need to count the number of ways we can select 4 marbles out of the 15 marbles. This can be done using the concept of combinations.
The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
Where:
- n is the total number of items
- r is the number of items to be chosen
- ! denotes the factorial operation (e.g., 4! = 4 x 3 x 2 x 1)
In our case, we want to select 4 marbles out of 15, so we can calculate it using the combination formula:
C(15, 4) = 15! / (4!(15-4)!)
C(15, 4) = (15 x 14 x 13 x 12) / (4 x 3 x 2 x 1)
C(15, 4) = 32,760 / 24
C(15, 4) = 1,365
Therefore, there are 1,365 sets of four marbles that include all the red ones.