From all the English alphabets, five letters are chosen and are arranged in alphabetical o…

Question

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 :

Accepted Answer

First, note that we are choosing 5 distinct letters (in strictly increasing alphabetical order) such that the middle (third) letter is `M'. Symbolically, if we denote the chosen letters as:

${L}_{1}we want ${L}\_{3}=$. The English alphabet has 26 letters, and M is the ${13}\^{}$.

Letters before : 1: Letters before M

The letters before M are ${A,B,C,\ldots ,L}$.

There are 12 letters here ($A$ through $L$).

We need to pick 2 of these 12 letters to occupy ${L}\_{1}$ and ${L}\_{2}$.

The number of ways to choose 2 letters out of 12 is ${}\^{12}{C}\_{2}$.

Letters after : 2: Letters after M

The letters after M are ${N,O,P,\ldots ,Z}$.

There are 13 letters here ($N$ through $Z$).

We need to pick 2 of these 13 letters to occupy ${L}\_{4}$ and ${L}\_{5}$.

The number of ways to choose 2 letters out of 13 is ${}\^{13}{C}\_{2}$.

Multiply the choices: Since these choices are independent (picking the two letters before M and two letters after M), the total number of ways is:

${}\^{12}{C}\_{2}\, 	imes \, {}\^{13}{C}\_{2}$

Calculate each combination:

${}\^{12}{C}\_{2}=\frac{12	imes 11}{2}=66,\, {}\^{13}{C}\_{2}=\frac{13	imes 12}{2}=78.$

So,

$

Solutions · 1