Question

There are $ 5 $ letters and $ 5 $ directed envelopes. The number of ways in which all the letters can be put in wrong envelope is
A. $ 119 $
B. $ 44 $
C. $ 59 $
D. $ 40 $

Answer 1

Hint: Here we have to find that no letter goes into the right envelope. For this, find the total ways in which all the letters go in the right envelope. That is only one letter can go in one correct envelope and subtract all of them from the total number of ways without any conditions irrespective of looking for right and wrong address.

Complete step-by-step answer:
There are $ 5 $ letters and $ 5 $ directed envelopes.
Therefore the total number of ways of posting letters is $ = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $ ways.
Also, the number of ways in which all the letters are in the correct directed envelope is $ = 1 $
The number of ways in which only $ 1 $ letter can go in $ 1 $ correct envelope, that is it can go in $ {}^5{C_1} $ ways and apparently other $ 4 $ letters can go wrong in $ 9 $ ways
 $ = {}^5{C_1} \times 9 = 5 \times 9 = 45 $ ways.
The number of ways in which only $ 2 $ letters can go in $ 2 $ correct envelopes, that is it can go in $ {}^5{C_2} $ ways and apparently other $ 3 $ letters can go wrong in $ 2 $ ways $ = {}^5{C_2} \times 2 = \dfrac{{5 \times 4}}{2} \times 2 = 20 $ ways.
The number of ways in which only $ 3 $ letter can go in $ 3 $ correct envelope, that is it can go in $ {}^5{C_3} $ ways and apparently other $ 2 $ letters can go wrong in $ 1 $ way $ = {}^5{C_3} \times 1 = \dfrac{{5 \times 4 \times 3}}{{3 \times 2}} \times 1 = 10 $ ways.
The number of ways in which only $ 4 $ letters can go in $ 4 $ correct envelope, that is it can go in zero ways since if four letters goes in the correct envelope then the fifth letter will also go in the correct envelope.
Number of ways $ = 0 $
Now, the total number of ways in which letters can be put in wrong envelopes is
 $ = 120 - 1 - 45 - 20 - 10 - 0 $
Simplify the above equations –
 $ = 44 $ ways
Hence, from the given multiple choices- the option B is the correct answer.
So, the correct answer is “Option B”.

Note: This question can also be solved by using the basic method of counting. But the use of the permutations and combinations gives us the efficient and accurate answer. Know the difference between the permutations and combinations formula and apply them accordingly.