5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes? Select:
A) 1 / 3
B) 2 / 3
C) 11 / 120
D) 11 / 30
E) 76 / 120

Rationale: this q is based on derangement , a permutation in which all elements are in the wrong position . number of derangements = n ! ( 12 ! − 13 ! + 14 ! + . . . + ( ( − 1 ) n ) n ! ) n ! ( 12 ! − 13 ! + 14 ! + . . . + ( ( − 1 ) n ) n ! ) . . of course it can be derived but thats not required . . since there are 5 letters and 5 envelopes : - derangements = 5 ! ( 12 ! − 13 ! + 14 ! − 15 ! ) derangements = 5 ! ( 12 ! − 13 ! + 14 ! − 15 ! ) = > 120 ( 1 / 2 − 1 / 6 + 1 / 24 − 1 / 120 ) = 60 − 20 + 5 − 1 = 44.120 ( 1 / 2 − 1 / 6 + 1 / 24 − 1 / 120 ) = 60 − 20 + 5 − 1 = 44 . total possible ways = 5 ! = 120 . therefore required prob , p = 44 / 120 = 11 / 30 . ans d
Answer: D