A secretary has 5 letters to send While he was away for lunc

A secretary has 5 letters to send. While he was away for lunch, his boss accidentally dropped all the envelopes and put them back on the tray in disorder. When the secretary comes back, not knowing that the envelopes have been reshuffled, he stuffs them. This results in letters going into envelopes at random.

(a) Define the sample space as a list of the form (L₁, . . . , L₅) where L_i gives the label of the envelope where the i-th letter has been put in. What is the cardinality of the sample space?

(b) Let T(n) be the number of ways in which n letters can all be put in n wrong envelopes (non of them goes in the correct one). Find T(2) and T(3). Then, show that T(n), n 3 satisfies the recursion: T(n) = (n 1)(T(n 1) + T(n 2)), and solve for n = 5.

(c) Assuming equiprobable outcomes, find the probability that at least one letter goes in the correct envelope.

Solution

(a) Sample space of an experiment is the set of all possible outcomes. In this the sample space is (L₁,L₂,L₃, L_{4 ,}L₅)

  the cardinality of the sample space is 5.

(b) Let T(n) be the number of ways in which n letters can all be put in n wrong envelopes

   T(1) be the number of ways in which 1 letters can all be put in 1 wrong envelopes=0

   T(2) be the number of ways in which 2 letters can all be put in 2 wrong envelopes=1

T(3) be the number of ways in which 3 letters can all be put in 3 wrong envelopes=2

that is written on T(2) and T(1) form

T(3)=2=2(T(1)+T(2))

for n terms

  T(n) = (n 1)(T(n 1) + T(n 2))

n=5

T(5) = 4(T(4) + T(3))

So,

T(4) be the number of ways in which 4 letters can all be put in 4 wrong envelopes=9

T(5)=4(9+2)=44