Permutations & Combinations
Grade None
Question:
<p>Consider all possible permutations of the letters of the word ENDEANOEL. The number of permutations in which letters A, E, O occur only in odd positions, is</p>
<p style="display:inline">2 <span class="math-tex">\(\times\)</span> 5!</p>
<p style="display:inline">5!</p>
<p style="display:inline">7 <span class="math-tex">\(\times\)</span> 5!</p>
<p style="display:inline">21 <span class="math-tex">\(\times\)</span> 5!</p>
Step-by-Step Solution
Key Concept: Divide the arrangements into independent tasks for odd and even positions and apply the formula for permutations of identical objects to each.
<p>There are five odd and 4 even positions. Five odd positions can be filled with A, E, E, E, O in <span class="math-tex">$\frac{5 !}{3 !}$</span> ways and four even positions can filled with D, N, N, L in <span class="math-tex">$\frac{4 !}{2 !}$</span> ways<br />
<span class="math-tex">$\Rightarrow $</span> The required number of permutations<br />
<span class="math-tex">$=\frac{5 !}{3 !} \times \frac{4 !}{2 !}$</span> = 2 <span class="math-tex">$\times$</span> 5!</p>
Correct Answer: A