Question

The number of ways of arranging 2 red, 3 white, and 5 yellow roses of different sizes into a garland such that no two yellow roses come together is:

• A. $2880$
• B. $144$
• C. $1440$
• D. $288$

Hint:

When arranging objects in a circle, use $(n-1)!$ for circular permutations, and for gap-based placement, use factorial permutations.

Answer 1

The Correct Option is C

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 $2 + 3 + 5 = 10$ roses. Step 2: Treat the garland as a circular arrangement In a circular arrangement (garland), the number of distinct arrangements of $n$ objects is \( (n
-1)! \). However, since the garland can be flipped, we divide by 2 to account for rotational symmetry. Thus, the total number of distinct arrangements is: \[ \frac{(n
-1)!}{2}. \] 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: \[ \frac{(5
-1)!}{2} = \frac{4!}{2} = 12. \] Step 4: Place the remaining roses After placing the 5 yellow roses, there are \( 10
- 5 = 5 \) positions left for the 2 red and 3 white roses. These 5 roses can be arranged in: $$\frac{5!}{2! \cdot 3!} = \frac{120}{2 \cdot 6} = 10$$ 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: $$12 \times 10 = 120.$$ 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 $2 + 3 = 5$ non
-yellow roses. In a circular arrangement, the number of distinct arrangements is: \[ \frac{(5
-1)!}{2} = \frac{24}{2} = 12. \] 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: $$5! = 120$$ 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: $$12 \times 120 = 1440.$$ Final Answer: $$\boxed{1440}$$