Permutations and Combinations

To prepare a garland using 6 distinct white roses and 6 distinct red roses such that no two red roses are adjacent, follow these steps:
Arranging White Roses:
Since we have 6 distinct white roses, they can be arranged in a garland. The formula to arrange flowers in a garland using circular permutations is ‌

(n−1)!

2

, where n is the number of items. For the white roses, this gives us:
‌

(6−1)!

2

=‌

5!

2

=‌

120

2

=60
Arranging Red Roses:
Placing Red Roses:
The number of ways to arrange these 6 distinct red roses in these 6 positions is given by the permutation of 6 distinct items, which is 6 !. Thus, the calculation is:
6!=720
Calculating Total Arrangements:
Finally, combine the permutations of both sets of roses. The total number of ways to arrange the garland while maintaining the condition is:
60×720=43200
Thus, there are 43,200 different ways to prepare the garland with the given conditions.