Permutations & Combinations
Grade 11
Question:
<p>How many 4-letter words can be made from the letters of the word <strong>MATHEMATICS</strong>?</p>
<p style="display:inline">2454</p>
<p style="display:inline">1680</p>
<p style="display:inline">1654</p>
<p style="display:inline">756</p>
Step-by-Step Solution
Key Concept: Partition the word formation into mutually exclusive cases based on the number of repeated letters and calculate the permutations for each case separately.
<p>There are 3 possibilities to make 4-letter words.<br />
<strong>Case I:</strong> All letters are distinct.<br />
There are 8 letters of different kinds.<br />
Number of 4 letter words = <sup>8</sup>P<sub>4</sub> = 1680<br />
<strong>Case II:</strong> Two distinct and 2 alike letters.<br />
There are 3 pairs of letters (MM, AA, TT), so we can choose one pair in <sup>3</sup>C<sub>1</sub> = 3 ways<br />
Now, 2 distinct letters can be chosen from the remaining 7 distinct letters, so the number of ways = <sup>7</sup>C<sub>2</sub> = 21 ways<br />
Again 4 letters can be arranged in <span class="math-tex">\(\frac{4 !}{2 !}\)</span> = 12 ways<br />
because one letter occurs twice<br />
Number of such words = 3 <span class="math-tex">\(\times\)</span> 2! <span class="math-tex">\(\times\)</span> 12 = 756<br />
<strong>Case III:</strong> Two pairs of alike letters.<br />
Since there are 3 pairs of alike letters in the given word (MM, AA, TT)<br />
Number of ways to choose 2 pairs = <sup>3</sup>C<sub>2</sub> = 3<br />
Again these 4-letters can be arranged in <span class="math-tex">\(\frac{4 !}{2 ! 2 !}\)</span> ways = 6 ways<br />
Number of arrangements of two pairs of alike letters = 3 <span class="math-tex">\(\times\)</span> 16 = 18<br />
<span class="math-tex">\(\Rightarrow\)</span> Required number of words = 1680 + 756 +18 = 2454</p>
Correct Answer: A