Question

In how many ways can we form a garland using 6 different flowers?

• A. 720
• B. 120
• C. 60
• D. 6

Hint:

In circular arrangements, use \((n-1)!\) for rotations; for garlands, divide further by 2 to account for reflections.

Answer 1

The Correct Option is C

Arranging objects in a garland (or necklace) is different from a simple permutation because:
1) Rotations of the arrangement are considered identical.
2) Reflections (flipping) are also considered identical.
The formula for the number of distinct arrangements of \(n\) distinct items in a garland is:
\[ \frac(n - 1)!2 \] Here, \(n = 6\).
So:
\[ \frac(6 - 1)!2 = \frac5!2 = \frac1202 = 60 \] Thus, there are \(60\) distinct garland arrangements possible.
Option (A), 720, is \(6!\), which counts all permutations without accounting for rotation and reflection equivalence.
Option (B), 120, is \(5!\), which accounts for rotation but not reflection.
Option (D), 6, is far too small, as it ignores most permutations.
Hence, the correct answer is 60.