Q1:The teacher of a class having a large number of students decides to take a test.After the test ended the teacher took the papers from the students and randomly distributed them among the students to check.
Prove that the probability that none of the students got their own paper to check is approximately = 1/e.
[Hint:Think of derangement]
Q2:There are m teachers and n children. Each teacher
gives one (random) child a cookie.
Prove that the probability that all n children get cookies is:
Σ(-1)k * nCk*(1-k/m)m.
[Hint:Think of Surjection]
591Answer to Q.1
Total number of ways in which no student will get his own paper = ∞![1/2! - 1/3! + 1/4! +.......upto infinity]
Total number of ways of getting the paper = ∞!
therefore probability = [1/2! - 1/3! + ....upto ∞]
which is equal to e^-1 = 1/e :)
Shaswata Roy Good one.
Akshay Ginodia :)