Thread: Putting letters into envelopes

A secretary randomly put 10 letters in 10 envelope
What is the probability that at least one letter was put in the correct envelope?

I solved it as follows

The probability of 0 letters in the correct envelope is
9/10 8/9.... = 9!/10!

what is my mistake ?

Originally Posted by Hitman6267

A secretary randomly put 10 letters in 10 envelope
What is the probability that at least one letter was put in the correct envelope?

You want a derangement of ten items.

I didn't understand how to apply the concept described in the wikipedia page.

Suppose we arrange the envelops in strict alphabetical order.
Now randomly order the letters. There are ways to do that.
From that reference there are arrangements where no letter is in its correct alphabetical position. You are asked to find the probability that at least one is in the correct position.

.

Is there any other way to resolve this ? Because we have never used such a formula.

Originally Posted by Hitman6267

Is there any other way to resolve this ? Because we have never used such a formula.

No other way that I am aware of.
That is a simple application of the inclusion/exclusion rule on ten objects.
I don’t understand why you were asked to do this problem without having been given the tools necessary to solve it.
Have you done the inclusion/exclusion rule?

Yes I have been taught the inclusion/exclusion rule
P(A U B U C)= P(A) + P(B) + P(C) - P(AC) - P(AB) - P(BC) + P(ABC)

where does that come in ?

Originally Posted by Hitman6267

Yes I have been taught the inclusion/exclusion rule
P(A U B U C)= P(A) + P(B) + P(C) - P(AC) - P(AB) - P(BC) + P(ABC)

Well use it for ten letters not just three.
A means that letter A is in the correct envelope.