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### Identical/Non-identical Groups; Derangements

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Date: 03/10/2000 at 04:54:27
From: Shweta Kala
Subject: Permutations and Combinations

Sir,

I was not able to understand the following:

1) The number of ways of dividing mp things into m identical groups of
p things each

= (mp)!/[(m!)(p!)^m]

2) The number of ways of dividing mp things into m different,
non-identical groups of p things each

= (mp)!/(p!)^m

3) Derangements: If n things are arranged in a row, the number of ways
in which they can be deranged so that no one of them occupies its
original place

= n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n.1/n!)

Why is there a different formula for derangements? Why isn't it

= P(n,n) - 1

After all, it is the number of arrangements of n things except the
original arrangement.

Kindly help me with this. Thank you,
Shweta Kala
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Date: 03/10/2000 at 07:23:04
From: Doctor Anthony
Subject: Re: Permutations and Combinations

1) In this situation the 'containers' of the groups are
indistinguishable, so you could swap the complete groups around in
m! ways without giving rise to a different combination. An example
would be putting p people into car(1), p people into identical
car(2), p people into identical car(3) and so on. We are only
concerned with the groups of people, and NOT worried about which
car they are in.

2) Here the 'containers' of the groups are different, e.g. you may be
putting p people into one house, another p people into a local
hotel, another p people into tents, and so on.

3) Derangements: P(n,n) = n! and (n! - 1)  would allow many of the
n objects to be in their original positions. For a derangement we
require EVERY single object to be in a different position from
where it was in the first arrangement.

A typical problem is that of the careless secretary who puts
letters at random into envelopes. We require the probability that
EVERY letter goes to a wrong address. For an explanation of this
problem, see:

Letters and Envelopes and the Inclusion-Exclusion Principle

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Permutations and Combinations

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