Question

There are 5 letters and 5 addressed envelopes. if the letters are put at random in the envelops, the probability that all the letters may be placed in wrongly addressed envelopes is

Answer 1

Hello..

If 1 letter and 1 envelope then then you can't put wrong (S1).

If 2 letters and 2 envelopes then 1 can be wrong (S2).

If 3 letters and 3 envelopes then 2 can be wrong (S3).

If 4 then you can put wrong in 9 ways(S4).

If 5 then it can put wrong in 44 ways(S5).

Now you can find a pattern.

S3=(S1+S2)*2

S4=(S2+S3)*3

S5=(S3+S4)*4

S6=(S4+S5)*5

= Sn=(Sn-2 + Sn-1)*(n-1)

if 5 letters then S5=(S3+S4)*4=(2+9)*4=44

Answer 2

Answer: $\dfrac{11}{30}$

Step-by-step explanation:

Given : There are 5 letters and 5 addressed envelopes.

The formula to find the number derangement is given by :-

$D_n=n!(1-\dfrac{1}{1!}+\dfrac{1}{2!}-\dfrac{1}{3!}+......+(-1)^n\dfrac{1}{n!}\ )$

For n= 5 , we have

$D_5=5!(1-\dfrac{1}{1!}+\dfrac{1}{2!}-\dfrac{1}{3!}+\dfrac{1}{4!}-\dfrac{1}{5!}\ )\\\\=(120)\times\dfrac{11}{30}=44$

Number of ways that all the letters may be placed in wrongly addressed envelopes is 44.

Total number of ways to put 5 letters in 5 envelope = 5!=120

Then, the probability that all the letters may be placed in wrongly addressed envelopes is $\dfrac{44}{120}=\dfrac{11}{30}$