A bag contains 5 red marbles, 3 green, 1 lavender, 3 yellow, 2 Orange.
Q. How many sets of four marbles include none of the red ones?
First I computed C(14,4) to find out how many sets of 4 could be formed. Total # sets =1001.
Then I subtracted the number of sets with 1, 2, 3, and 4 red marbles:
C(5,1) x C(9,3) = 84
C(5,2) x C(9,2) = 360
C(5,3) x C(9,1) = 180
C(5,4) = 5
Total sets with red marbles = 629
Final answer = 1001 - 629 = 372
Is this correct?
Yes, your method of calculating the number of sets of four marbles that do not include any of the red marbles is correct.
To explain in more detail:
First, you correctly computed C(14,4) to find out how many sets of 4 could be formed from the total of 14 marbles. The notation C(n,r), also known as a combination or binomial coefficient, represents the number of ways to choose r items from a set of n items without regard to the order.
Using the formula C(n,r) = n! / (r!(n-r)!), where n! represents the factorial of n, you obtained C(14,4) = 1001.
Next, you calculated the number of sets with 1, 2, 3, and 4 red marbles and subtracted them from the total number of sets. This ensures that only sets without any red marbles are counted.
For example, to find the number of sets with exactly 1 red marble, you computed C(5,1) x C(9,3). This represents choosing 1 red marble from the 5 available and choosing 3 marbles from the remaining 9 non-red marbles. Similar calculations were used to obtain the number of sets with 2, 3, and 4 red marbles.
By subtracting the total number of sets with red marbles (629) from the total number of sets possible (1001), you correctly obtained the final answer of 372 sets of four marbles that include none of the red ones.