Circular Permutation

To solve this problem, we need to arrange 2 red, 3 white, and 5 yellow roses of different sizes into a garland such that no two yellow roses come together. Here's the step
-by
-step solution: Step 1: Total number of roses There are a total of roses. Step 2: Treat the garland as a circular arrangement In a circular arrangement (garland), the number of distinct arrangements of objects is . However, since the garland can be flipped, we divide by 2 to account for rotational symmetry. Thus, the total number of distinct arrangements is: Step 3: Fix the yellow roses first To ensure that no two yellow roses are adjacent, we first place the yellow roses in the garland. Since there are 5 yellow roses, we place them in such a way that they are separated by the other roses. In a circular arrangement, the number of ways to place 5 yellow roses such that no two are adjacent is: Step 4: Place the remaining roses After placing the 5 yellow roses, there are positions left for the 2 red and 3 white roses. These 5 roses can be arranged in: ways, accounting for the indistinguishability of the red and white roses. Step 5: Total number of arrangements Multiply the number of ways to place the yellow roses by the number of ways to place the remaining roses: However, this does not match any of the options. Let's reconsider the problem. Step 6: Correct approach The correct approach is to treat the garland as a circular arrangement and use the gap method to ensure no two yellow roses are adjacent. 1. Arrange the non
-yellow roses: There are non
-yellow roses. In a circular arrangement, the number of distinct arrangements is: 2. Place the yellow roses: After arranging the 5 non
-yellow roses, there are 5 gaps between them where the yellow roses can be placed. We need to place 5 yellow roses into these 5 gaps such that no two yellow roses are in the same gap. This can be done in: ways. 3. Total arrangements: Multiply the number of ways to arrange the non
-yellow roses by the number of ways to place the yellow roses: Final Answer: