Question:
5 letters are randomly selected from English alphabets and they are arranged in alphabetical order. The probability that the 5 letters selected and arranged has \(M\) in the middle place is
5 letters are randomly selected from English alphabets and they are arranged in alphabetical order. The probability that the 5 letters selected and arranged has \(M\) in the middle place is
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When letters are arranged in alphabetical order, the middle position is determined by rank. For a letter to occupy the middle position among 5 selected letters, exactly two selected letters must be smaller and two must be larger than that letter.
Updated On: Jul 29, 2026
- \(\dfrac{9}{115}\)
- \(\dfrac{{}^{12}C_2\cdot{}^{13}C_2}{{}^{26}C_5}\)
- \(\dfrac{{}^{25}C_4}{{}^{26}C_5}\)
- \(\dfrac{9}{125}\)
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The Correct Option is A
Solution and Explanation
Concept:
Since the selected letters are arranged in alphabetical order, \(M\) will occupy the middle (third) position if exactly two selected letters are alphabetically smaller than \(M\) and exactly two are greater than \(M\).
Step 1: Count letters before and after \(M\). The English alphabet contains \(26\) letters. \(M\) is the \(13^{th}\) letter. Hence, \[ \text{Letters before }M=12, \] \[ \text{Letters after }M=13. \]
Step 2: Find the number of favourable selections. For \(M\) to be in the middle position after arranging alphabetically, \[ \text{Choose 2 letters from the 12 letters before }M, \] and \[ \text{Choose 2 letters from the 13 letters after }M. \] Therefore, \[ \text{Favourable selections} = {12 \choose 2}{13 \choose 2}. \] \[ = 66\times78 = 5148. \]
Step 3: Find the total number of selections. The total number of ways to select \(5\) letters from \(26\) letters is \[ {26 \choose 5}. \] \[ = 65780. \]
Step 4: Calculate the probability. \[ P = \frac{{12 \choose 2}{13 \choose 2}} {{26 \choose 5}}. \] \[ = \frac{5148}{65780}. \] \[ = \frac{9}{115}. \] Therefore, \[ \boxed{\frac{9}{115}} \] \[ \boxed{\text{Answer = (A)}} \]
Step 1: Count letters before and after \(M\). The English alphabet contains \(26\) letters. \(M\) is the \(13^{th}\) letter. Hence, \[ \text{Letters before }M=12, \] \[ \text{Letters after }M=13. \]
Step 2: Find the number of favourable selections. For \(M\) to be in the middle position after arranging alphabetically, \[ \text{Choose 2 letters from the 12 letters before }M, \] and \[ \text{Choose 2 letters from the 13 letters after }M. \] Therefore, \[ \text{Favourable selections} = {12 \choose 2}{13 \choose 2}. \] \[ = 66\times78 = 5148. \]
Step 3: Find the total number of selections. The total number of ways to select \(5\) letters from \(26\) letters is \[ {26 \choose 5}. \] \[ = 65780. \]
Step 4: Calculate the probability. \[ P = \frac{{12 \choose 2}{13 \choose 2}} {{26 \choose 5}}. \] \[ = \frac{5148}{65780}. \] \[ = \frac{9}{115}. \] Therefore, \[ \boxed{\frac{9}{115}} \] \[ \boxed{\text{Answer = (A)}} \]
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