A bag contains 3 red marbles, 2 green ones, 1 lavender one, 2 yellows, and 4 orange marbles. How many sets of four marbles include all the red ones?
1Your answer is incorrect.
If a set of 4 includes the 3 red marbles, it must also include one of the other colors, green, lavender, yellow or orange.
Does that help you understand the situation?
To determine how many sets of four marbles include all the red ones, we need to consider the total number of sets of four marbles that can be chosen.
First, let's find the total number of marbles in the bag:
3 (red) + 2 (green) + 1 (lavender) + 2 (yellow) + 4 (orange) = 12 marbles
To select four marbles from the bag, we can use the combination formula, which is represented as C(n, r), where n is the total number of items and r is the number of items we want to choose. In this case, n = 12 (total marbles) and r = 4 (marbles including all the red ones).
The formula for the combination is:
C(n, r) = n! / (r!(n-r)!)
Plugging in the values:
C(12, 4) = 12! / (4!(12-4)!) = 12! / (4!8!)
Next, we need to calculate the factorial of 12:
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Now, let's calculate the factorial of 4:
4! = 4 x 3 x 2 x 1
And the factorial of 8:
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Now we can substitute these values into the formula:
C(12, 4) = 12! / (4!8!) = (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((4 x 3 x 2 x 1) x (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1))
After simplifying the equation, we can find the total number of sets of four marbles that include all the red ones.