A bag contains 3 red marbles, 1 green one, 1 lavender one, 3 yellows, and 3 orange marbles. HINT [See Example 7.]
How many sets of five marbles include at least two red ones?
378
To find the number of sets of five marbles that include at least two red ones, we can break down the problem into several cases.
Case 1: Exactly 2 red marbles:
In this case, we need to choose 2 red marbles from the 3 available, and we can choose the remaining 3 marbles from any of the other colors. The number of ways to do this is given by the binomial coefficient (3C2) * (10C3).
Case 2: Exactly 3 red marbles:
In this case, we need to choose all 3 red marbles from the available 3, and we can choose the remaining 2 marbles from any of the other colors. The number of ways to do this is given by (3C3) * (10C2).
Case 3: 4 or 5 red marbles:
In these cases, we simply need to choose all 5 marbles from the available 3 red marbles, and we can choose the remaining 0 or 1 marbles from any of the other colors. The number of ways to do this is given by (3C5) + (3C4).
To get the total number of sets, we sum up the results of all these cases:
Total number of sets = (3C2) * (10C3) + (3C3) * (10C2) + (3C5) + (3C4)
Let's calculate the answer:
(3C2) = 3 (10C3) = 120
(3C3) = 1 (10C2) = 45
(3C5) = 0 (3C4) = 0
Total number of sets = (3 * 120) + (1 * 45) + 0 + 0 = 360 + 45 = 405
Therefore, there are 405 sets of five marbles that include at least two red ones.