Permutations & Combinations
Grade None
Question:
<p>The number of ways in which 5 beads of different colours form a necklace is</p>
<p style="display:inline">12</p>
<p style="display:inline">24</p>
<p style="display:inline">120</p>
<p style="display:inline">60</p>
Step-by-Step Solution
Key Concept: The number of ways to arrange n distinct beads in a necklace is (n-1)! / 2 because clockwise and anticlockwise circular arrangements are identical when the necklace can be flipped.
<p>The number of ways in which 5 beads of different colours can be arranged in a circle to form a necklace are (5 - 1)! = 4! = 24<br />
Since the clockwise and anticlockwise arrangement is the same.<br />
<span class="math-tex">$\Rightarrow$</span> Total number of ways of arranging the beads<br />
= <span class="math-tex">$\frac{1}{2} \times$</span> 4! = <span class="math-tex">$\frac{1}{2} \times$</span> 4 <span class="math-tex">$\times$</span> 3 <span class="math-tex">$\times$</span> 2 = 12</p>
Correct Answer: A