Question

The number of different ways of preparing a garland using 6 distinct white roses and 6 distinct red roses such that no two red roses come together is:

• A. \( 43200 \)
• B. \( 86400 \)
• C. \( 59200 \)
• D. \( 76800 \)

Hint:

For circular permutations, fix one object to avoid identical rotations, and arrange the remaining \( n-1 \) objects normally.

Answer 1

The Correct Option is A

Step 1: Arranging the White Roses in a Circular Pattern - Since the garland is circular, one white rose is fixed to eliminate equivalent rotations. - The remaining 5 white roses can be arranged in: \[ (6-1)! = 5! = 120 \] Step 2: Identifying Slots for Red Roses - Once the white roses are arranged, they form 6 distinct gaps where red roses can be placed. - Since we must ensure that no two red roses are adjacent, each red rose must occupy a separate gap. Step 3: Arranging the Red Roses in These Gaps - The 6 distinct red roses can be arranged among themselves in: \[ 6! = 720 \] Step 4: Computing the Total Arrangements - The final number of ways to form the garland while maintaining the given condition is: \[ (6-1)! \times 6! = \frac{ 5! \times 6! } {2}= 43200 \] Thus, the correct answer is 43200.