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Dear Raman,
Any change in the existing order of things is called a derangement.
If ''n'' things are arranged in a row, the number of ways in which they can, be deranged so that none of them occupies its original place is
and it is denoted by D(n).
A question on derangement can be of the following kind:
Illustration:
Supposing 4 letters are placed in 4 different envelopes. In how many ways can be they be taken out from their original envelopes and distributed among the 4 different envelopes so that no letter remains in its original envelope?
Solution:
Using the formula for the number of derangements that are possible out of 4 letters in 4 envelopes, we get the number of ways as :
4!(1 - 1 + 1/2! - 1/3! + 1/4!) = 24(1 - 1 + 1/2 - 1/6 + 1/24) = 9.
Let x1, x2, ......, xm be integers. Then number of solutions to the equation
x1 + x2 + ... + xm = n ... (1)
subject to the conditions a1 < x1 < b1, a2 < x2 < b2, ..., am < xm < bm ... (2)
is equal to the coefficient of xn in
...(3)
This is because the number of ways in which sum of m integers in (1) subject to given conditions (2) equals n is the same as the number of times xn comes in (3). Using this we get the number of non negative integral solutions of (1) is given byn+m-1Cm-1 and number of positive integral solutions of (1) is given by n-1Cm-1.
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Multinomial theorem is used when you need to do selection from alike objects.
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