Permutations & Combinations
Grade None
Question:
<p>The number of ways in which 4 boys and 4 girls can stand in a circle so that each boy and each girl is one after the other is :</p>
<p style="display:inline"><span class="math-tex">\(4 ! \cdot 4 !\)</span></p>
<p style="display:inline"><span class="math-tex">\(3 ! 4 !\)</span></p>
<p style="display:inline"><span class="math-tex">\(8 !\)</span></p>
<p style="display:inline"><span class="math-tex">\(7 !\)</span></p>
Step-by-Step Solution
Key Concept: To arrange two equal groups alternately in a circle, fix one group using the circular permutation formula (n-1)! and then place the second group in the gaps using the linear permutation formula n!.
<p>First average boys in a circle<br />
<span class="math-tex">\(\Rightarrow\)</span> (4-1)! = 3!<br />
Now,<br />
in the 4 spaces generalised between boys = Arrange girls,<br />
<span class="math-tex">\(\therefore\)</span> number of ways<br />
= <span class="math-tex">\( 3 ! \cdot 4 !\)</span> <span class="math-tex">\(\rightarrow\)</span> linear arrangement<br />
= 6 <span class="math-tex">\(\times\)</span> 24<br />
= 144</p>
Correct Answer: B