Question

A person writes $4$ letters and addresses $4$ envelopes. If the letters are placed in the envelopes at random, what is the probability that all letters are not placed in the right envelopes?
A. $\dfrac{1}{4}$
B. $\dfrac{{11}}{{24}}$
C. $\dfrac{{15}}{{24}}$
D. $\dfrac{{23}}{{24}}$

Answer 1

Hint:
The number of ways in which $4$ envelopes can be placed in the $4$ envelopes $ = 4! = 24$
And if all the letters go into the right envelopes so that can be done only in $1$ way
So the probability of getting all the letters in the right envelopes $ = \dfrac{1}{{24}}$
Hence we can also find the probability of not placing all the letters in the right envelopes.

Complete step by step solution:
Here we are given in the question that person writes $4$ letters and addresses $4$ envelopes. Let us assume that the four letters he wrote be represented by ${L_1},{L_2},{L_3},{L_4}$ and the envelopes be ${E_1},{E_2},{E_3},{E_4}$ respectively. If ${L_1}$ goes into${E_1}$, ${L_2}$ goes into${E_2}$, ${L_3}$goes into${E_3}$, ${L_4}$ goes into the envelope ${E_4}$ then only we can say that all the letters go into the right envelopes and hence there is only one possible way of that.
Also we know that total number of ways in which four letters can be put into the four envelopes$ = 4!$$ = 24$
As we know that ${}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$
So ${}^4{C_4} = \dfrac{{4!}}{{(4 - 4)!4!}}$ and we also know that $0! = 1$
Hence we get that ${}^4{C_4} = \dfrac{{4!}}{{0!4!}} = 1$
So there is only one possible way to put the $4$ letters in $4$ envelopes.
Also we know that total number of ways in which four letters can be put into the four envelopes $ = 4!$$ = 24$
So the probability of letter going in the right envelop is
$P(E) = \dfrac{{{\text{number of favourable outcomes }}}}{{{\text{total number of conditions}}}} = \dfrac{1}{{24}}$
But we want the probability of letter not going into the right envelope so that will be
$
  1 - P(E) \\ \Rightarrow
  1 - \dfrac{1}{{24}} \\ \Rightarrow
  \dfrac{{23}}{{24}} \\
 $

Note:
${}^n{C_r}$ refers to the number of ways in which r number of items can be chosen from n number of different items and also we know that ${}^n{C_r} = {}^n{C_{n - r}}$ and also ${}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$