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A postman has to deliver fi ve letters to five di fferent houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many di fferent ways could he do this so that exactly two of the five houses receive the correct letters?

A postman has to deliver fi ve letters to five di fferent houses. Mischievously, he posts one
letter through each door without looking to see if it is the correct address. In how many
di fferent ways could he do this so that exactly two of the five houses receive the correct
letters?

Grade:

1 Answers

Aman Bansal
592 Points
8 years ago

Dear Student,

Suppose Houses are ABCDE and corresponding letters are ABCDE. 

So the total no of correct probabilities are :

ABECD, AECBD and so on.. 
And total no of possibilities are :5!

 

A derangement is a permutation in which none of the objects appear in their "natural" (i.e., ordered) place. For example, the only derangements of {1,2,3} are {2,3,1} and {3,1,2}, so !3=2. Similarly, the derangements of {1,2,3,4} are {2,1,4,3}{2,3,4,1}{2,4,1,3}{3,1,4,2}{3,4,1,2}{3,4,2,1}{4,1,2,3}{4,3,1,2}, and {4,3,2,1}. Derangements are permutations without fixed points (i.e., having no cycles of length one). The derangements of a list of n elements can be computed using Derangements[n] in the Mathematica package Combinatorica` .

The problem was formulated by P. R. de Montmort in 1708, and solved by him in 1713 (de Montmort 1713-1714). Nicholas Bernoulli also solved the problem using the inclusion-exclusion principle (de Montmort 1713-1714, p. 301; Bhatnagar 1995, p. 8).

Derangements are also called rencontres numbers (named after rencontres solitaire) or complete permutations, and the number of derangements !non n elements is called the subfactorial of n.

The function giving the number of distinct derangements on n elements is called the subfactorial !n and is equal to

!n = n!sum_(k=0)^(n)((-1)^k)/(k!)
(1)
= (Gamma(n+1,-1))/e
(2)

(Bhatnagar 1995, pp. 8-9), where Gamma(z,a) is the incomplete gamma function, or

 !n=[(n!)/e],
(3)

where k! is the usual factorial and [x] is the nearest integer function.

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Thanks

Aman Bansal

Askiitian Expert

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