Final answer:
The number of ways in which 5 letters can be placed into 5 addressed envelopes so that none of the letters goes into the correct envelope is 44, which is a derangement problem denoted by !5.
Step-by-step explanation:
The problem is to determine the number of ways in which 5 letters can be placed into 5 addressed envelopes so that none of the letters goes into its corresponding envelope. This is a problem of derangements, also known as a permutation where no element appears in its original position. The derangement of n objects is denoted by !n or D(n). The formula for derangement is given by:
!n = n! ∑ ((-1)^k/k!), where the sum is taken from k=0 to n.
For n=5, the calculation goes as follows:
!5 = 5! ∑ ((-1)^k/k!)
= 5! ((-1)^0/0! + (-1)^1/1! + (-1)^2/2! + (-1)^3/3! + (-1)^4/4! + (-1)^5/5!)
= 120 (1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120 (1 - 1 + 0.5 - 0.1666 + 0.041666 - 0.008333)
= 120 (0.366833)
= 44
Thus, the answer is B. 44.