Combination Analysis of Bubble Gum Selections
When exploring selection problems, such as choosing bubble gums, we often need to apply combinatorial principles. In this article, we will demonstrate how to determine the number of possible selections of five bubble gums, given that we must select two of each of three colors. We will break down the problem step by step and use combinatorial formulas to find the solution.
Problem Statement
The problem states that we have a total of 3 red, 4 green, and 5 yellow bubble gums. We need to find out how many different selections of five bubble gums are possible such that we select exactly two gum pieces of each color. This means we are looking for combinations where we select 2 out of 3 red, 2 out of 4 green, and 2 out of 5 yellow pieces.
Step-by-Step Solution
Step 1: Calculate Combinations for Each Color
We will use the combination formula, denoted as ( C(n, k) ), which is the number of ways to choose k items from n items without regard to the order of selection.
Red:
We need to choose 2 out of 3 available red bubble gums.
C(3, 2) (frac{3!}{2!(3-2)!} frac{3 times 2 times 1}{2 times 1 times 1} 3)
Green:
We need to choose 2 out of 4 available green bubble gums.
C(4, 2) (frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6)
Yellow:
We need to choose 2 out of 5 available yellow bubble gums.
C(5, 2) (frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10)
Step 2: Multiply the Combinations
To find the total number of ways to make the selection, we multiply the combinations for each color.
Total selections C(3, 2) × C(4, 2) × C(5, 2) 3 × 6 × 10
3 × 6 18
18 × 10 180
Conclusion
The total number of selections consisting of five bubble gums, each with two of each color, is 180.
Alternative Approach
Alternatively, we can assume different distribution scenarios, such as selecting 2 of each color with one additional piece of another color. For example, the combinations can be 2-2-1, 2-1-2, 1-2-2, etc. Here are the calculations for these scenarios:
X2, Y2, Z1: C(3, 2) × C(4, 2) × C(5, 1) 3 × 6 × 5 92, Y1, Z2: C(3, 2) × C(4, 1) × C(5, 2) 3 × 4 × 10 121, Y2, Z2: C(3, 1) × C(4, 2) × C(5, 2) 3 × 6 × 10 180Summing these up: 90 120 180 390.
Final Answer
The total number of selections is either 180 or 390, depending on the distribution of the fifth bubble gum.