Question

There are five letters and five addressed envelopes. If the letters are placed at random, the probability that exactly three letters are placed in right envelope is:
(A) $\dfrac{3}{5}$
(B) $\dfrac{1}{6}$
(C) $\dfrac{1}{{12}}$
(D) $\dfrac{1}{3}$

Answer 1

In the given question we are provided with five letters and five addressed envelopes. We are required to find the probability of exactly three letters placed in the right envelopes. So, two of the five letters are to be placed in wrong envelopes. So, we have to choose three out of five letters that are to be placed wrongly and then arrange all the letters. We will make use of the combination formula $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ to find the number of ways in which exactly three letters go in right envelopes and then find the probability using the formula $\dfrac{{Favourable\,number\,of\,outcomes}}{{Total\,number\,of\,outcomes}}$.

Complete answer:
So, we have a total of five letters to be placed in five envelopes.
Hence, the ways of arranging n things is equal to $n!$.
Therefore, the total number of ways of placing five letters in five envelopes is $5! = 120$.
Now, we have to calculate the number of ways in which three out of the five letters are placed in the right envelopes.
So, the number of ways of selecting three out of five letters that will be placed in the right envelope is $^5{C_3} = 10$.
We know that there is only one right envelope corresponding to each letter. So, there is a single way of selecting a right envelope for all the three letters that are correctly placed. Also, the remaining two letters that are to be placed in wrong envelopes have only one option of being placed in each other’s envelope.
So, we get a favorable number of ways as $10$.
Now, we calculate the probability using the formula $\dfrac{{Favourable\,number\,of\,outcomes}}{{Total\,number\,of\,outcomes}}$ and substituting the values of number of total and favorable ways.
So, required probability $= \dfrac{{10}}{{120}} = \dfrac{1}{{12}}$
Hence, the probability of placing exactly three letters out of five in the right envelopes is $\dfrac{1}{{12}}$.
Hence, option (C) is correct.

Note:
The question revolves around the concepts of probability and permutations and combinations. One should know about the principle rule of counting or the multiplication rule. Care should be taken while handling the calculations. Calculations should be verified once so as to be sure of the answer. We must know the combination formula $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ to calculate the number of ways of selecting r things out of n different things.