Unique Arrangements of Rosebushes: A Combinatorial Problem Explained

Combinatorial problems often challenge us to find the number of ways elements can be arranged under specific conditions. In this article, we'll explore a classic problem involving Mr. Flowers and his 10 rosebushes. The challenge is to determine the number of distinct ways Mr. Flowers can plant his rosebushes in a row, given that six of the bushes are white, two are yellow, and two are red.

Introduction to the Problem

Mr. Flowers has 10 rosebushes in total, consisting of six white, two yellow, and two red bushes. The task is to find the number of unique ways he can plant these bushes in a row. To solve this, we need to consider permutations and combinations, given that some of the bushes are indistinguishable within their color groups.

Using Permutations and Combinations

The total number of ways to arrange 10 distinct items is given by the factorial of 10, or 10!. However, since the white rosebushes, yellow rosebushes, and red rosebushes are indistinguishable within their own groups, we need to account for this by dividing by the factorials of the counts of these colors. Thus, the formula becomes:

10! / (6! times; 2! times; 2!)

Step-by-Step Calculation

Let's break down the calculation step-by-step:

Calculate 10! (factorial of 10): Calculate 6! (factorial of 6): Calculate 2! (factorial of 2) for both the yellow and red rosebushes: Substitute these values into the formula 10! / (6! times; 2! times; 2!): Perform the division to get the number of distinct arrangements:

Coded Calculation

Using the values, we have:

10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 3,628,800

6! 6 x 5 x 4 x 3 x 2 x 1 720

2! 2 x 1 2

So, the formula becomes:

3,628,800 / (720 times; 2 times; 2) 3,628,800 / 2,880 1,260

Alternative Approach

Another way to solve this problem is to use the concept of choosing groups of plants. We can break down the calculation as follows:

First, choose 6 out of 10 for the white rosebushes. The number of ways to do this is given by the combination formula C(10, 6) 10! / (4! times; 6!): Then, choose 2 out of the remaining 4 for the yellow rosebushes. The number of ways to do this is given by the combination formula C(4, 2) 4! / (2! times; 2!): The remaining 2 will automatically be the red rosebushes. There's only 1 way to choose these 2 red rosebushes from 2.

The total number of unique arrangements is the product of these combinations:

10C6 times; 4C2 times; 2C2 (10! / (4! times; 6!)) times; (4! / (2! times; 2!)) times; 1

Substituting in the factorial values, we get:

(3,628,800 / (24 times; 720)) times; (24 / (2 times; 2)) times; 1 210 times; 6 1,260

Conclusion

Therefore, the number of distinct ways Mr. Flowers can plant the 10 rosebushes, considering the constraints, is 1,260. This calculation showcases the power of combinatorial methods in solving problems where elements are not all distinct.

Key Takeaways

Combinatorics is a key tool in solving problems involving arrangements with constraints. Permutations and combinations help in accurately counting unique arrangements. Factorials and combinatorial formulas are used to handle indistinguishable elements.

Frequently Asked Questions

What is the difference between permutations and combinations?
Permutations are used when the order of elements matters, while combinations are used when the order does not matter. Why are we dividing by the factorials of the groups?
We divide by the factorials of the groups to avoid overcounting, as some elements within a group are indistinguishable. Can this method be applied to other similar problems?
Yes, this method can be applied to similar problems involving the arrangement of indistinguishable and distinguishable elements.