<p>The number of four-letter words that can be formed using the letters of the word <strong>BARRACK </strong>is</p>
<p style="display:inline">270</p>
<p style="display:inline">120</p>
<p style="display:inline">264</p>
<p style="display:inline">144</p>
Step-by-Step Solution
Key Concept: Categorize the possible letter combinations into mutually exclusive cases based on repetition—all distinct, one pair of identical letters, or two pairs—and sum their permutations.
<p>If all four letters are different then the number of words <sup>5</sup>C<sub>4</sub> <span class="math-tex">\(\times\)</span> 4! = 120<br />
If two letters are R and other two different letters are chosen from B, A, C, K then the number of words = <sup>4</sup>C<sub>2</sub> <span class="math-tex">\(\times\)</span> <span class="math-tex">\(\frac{4 !}{2 !}\)</span> = 72<br />
If two letters are A and other two different letters are chosen from B, R, C, K then the number of words = <sup>4</sup>C<sub>2</sub> <span class="math-tex">\(\times\)</span> <span class="math-tex">\(\frac{4 !}{2 !}\)</span> = 72<br />
If a word is formed using two R’s and two A’s then the number of words = <span class="math-tex">\(\frac{4 !}{2 ! 2 !}\)</span> = 6<br />
Therefore, the number of four-letter words that can be formed = 120 + 72 + 72 + 6 = 270</p>
Correct Answer: A