Putting letters into envelopes

• Jul 11th 2010, 07:42 AM
Hitman6267
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 ?
• Jul 11th 2010, 08:05 AM
Plato
Quote:

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.
• Jul 11th 2010, 10:12 AM
Hitman6267
I didn't understand how to apply the concept described in the wikipedia page.
• Jul 11th 2010, 10:28 AM
Plato
Suppose we arrange the envelops in strict alphabetical order.
Now randomly order the letters. There are $10!$ ways to do that.
From that reference there are $d_{10}$ 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.

${\displaystyle 1-\frac{d_{10}}{10!}$.
• Jul 11th 2010, 10:32 AM
Hitman6267
Is there any other way to resolve this ? Because we have never used such a formula.
• Jul 11th 2010, 10:41 AM
Plato
Quote:

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?
• Jul 11th 2010, 10:45 AM
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)

where does that come in ?
• Jul 11th 2010, 12:15 PM
Plato
Quote:

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.