From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

<p>First, note that we are choosing <strong>5 distinct letters</strong> (in strictly increasing alphabetical order) such that the <strong>middle (third) letter is ‘M’</strong>. Symbolically, if we denote the chosen letters as:</p> <p>$ L_1 < L_2 < L_3 < L_4 < L_5, $</p> <p>we want $L_3 = \text{M}$. The English alphabet has 26 letters, and M is the $13^\text{th}$.</p> <hr /> <h3>Step 1: Letters before M</h3> <p><p>The letters before M are $\{A, B, C, \ldots, L\}$. </p></p> <p><p>There are <strong>12</strong> letters here ($A$ through $L$). </p></p> <p><p>We need to pick <strong>2</strong> of these 12 letters to occupy $L_1$ and $L_2$. </p></p> <p><p>The number of ways to choose 2 letters out of 12 is ${ }^{12} \mathrm{C}_2$.</p></p> <h3>Step 2: Letters after M</h3> <p><p>The letters after M are $\{N, O, P, \ldots, Z\}$. </p></p> <p><p>There are <strong>13</strong> letters here ($N$ through $Z$). </p></p> <p><p>We need to pick <strong>2</strong> of these 13 letters to occupy $L_4$ and $L_5$. </p></p> <p><p>The number of ways to choose 2 letters out of 13 is ${ }^{13} \mathrm{C}_2$.</p></p> <h3>Step 3: Multiply the choices</h3> <p>Since these choices are independent (picking the two letters before M and two letters after M), the total number of ways is:</p> <p>$ { }^{12} \mathrm{C}_2 \;\times\;{ }^{13} \mathrm{C}_2 $</p> <p>Calculate each combination:</p> <p>$ { }^{12} \mathrm{C}_2 = \frac{12 \times 11}{2} = 66, \quad { }^{13} \mathrm{C}_2 = \frac{13 \times 12}{2} = 78. $</p> <p>So,</p> <p>$ { }^{12} \mathrm{C}_2 \times { }^{13} \mathrm{C}_2 = 66 \times 78 = 5148. $</p> <hr /> <h2><strong>Answer: 5148</strong> (Option B)</h2>