# Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an a

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Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta.
There are 5 different colour balls and 5 boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is
A. 40
B. 44
C. 45
D. 60

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Correct Answer - B
(b) (D_(n))/(n!)-(D_(n-1))/((n-1)!)=((-1)^(n))/(n!) gives
(D_(n))/(n!)=sum_(r=2)^(n)((D_(r ))/(r!)-(D_(r-1))/((r-1)!))=sum_(r=2)^(n)((-1)^(r ))/(r!)
impliesD_(n)=n!sum_(r=2)^(n)((-1)^(r ))/(r!)
:. D_(5)=5!((1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!))=44

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