Question

There are 7 groups in a picnic who has brought their own lunch box, and then the 7 lunch box are exchanged within those groups. Determine the number of ways that they can exchange the lunch box such that none of them can get their own.

(a) 655

(b) 328

(c) 1854

(d) 3765

This question was posed to me in a national level competition.

I need to ask this question from Counting topic in section Counting of Discrete Mathematics

Answer 1

Correct choice is (c) 1854

For explanation: This can be solved by the derangement formula:

!n = n!(1 – \(\frac{1}{1!} + \frac{1}{2!} – \frac{1}{3!} + … + \frac{(-1)^n1}{n!}\)) ⇒ 7! = 1854.