Question

A person writes n letters and addresses on envelopes; if the letters are placed in the envelopes at random, what is the probability that every letter goes wrong?

Answer 1

Hint: Here we will proceed by assuming ${u_n}$as the required answer i.e. number of ways in which all the letters go wrong. Then we will find the required probability by formula - ${u_n} = \left( {n - 1} \right)\left( {{u_{n - 1}} + {u_{n - 2}}} \right)$.

Complete step-by-step answer:

Let ${u_n}$ deNote:the number of ways in which all the letters go wrong
Let a, b, c, d…. represent that arrangement in which all the letters are in their own envelopes.
Now if $a$ in any other arrangement occupies the place of an assigned letter $b$, this letter must either occupy a’s place or some other.

1. Suppose $b$ occupies a’s place. Then the number of ways in which all the remaining n-2 letters can be displaced is ${u_{n - 2}}$.
2. Therefore the numbers of ways in which $a$may be displaced by interchange with some one of the other n-1 letters, and the rest be all displaced is $\left( {n - 1} \right){u_{n - 1}}$.
3. Suppose $a$ occupies b’s place, and $b$ does not occupy a’s place. Then in arrangements satisfying the required conditions, since $a$is fixed in b’s place, the letter b, c, d, ……. must be all displaced, which can be done in ${u_{n - 1}}$ ways.
4. Therefore the number of ways in which $a$occupies the place of another letter but not by interchange with that letter is-
$\therefore$ ${u_n} = \left( {n - 1} \right)\left( {{u_{n - 1}} + {u_{n - 2}}} \right)$ ;
Now we will find ${u_n} - n{u_{n - 1}} = {\left( { - 1} \right)^n}\left( {{u_2} - {u_1}} \right)$
Also ${u_1} = 0,{u_2} = 1$
Thus, we finally obtain-
{u_n} = n\left\\{ {\dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{4} - ..... + \dfrac{{{{\left( { - 1} \right)}^n}}}{n}} \right\\}
Now the total number of ways in which the $n$ things can be put in $n$ places,
Therefore, the required chance is –
$\dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{4} - ..... + \dfrac{{{{\left( { - 1} \right)}^n}}}{n}$

Note: Here in this question, we can assume any variable instead of x to find the probability of every letter goes wrong. As a and b are our assumed variables for converting given conditions, we can use any other variables so that we can find the required answer using the given conditions.