![]() The answer is the number of ways the hats can be arranged so that there is no hat in its original position divided by n!, the number of permutations of n hats. What is the probability that no one receives the correct hat? Remark 1. When customers return for their hats, the checker gives them back hats chosen at random from the remaining hats. A new employee checks the hats of n people at a restaurant, forgetting to put claim check numbers on the hats. 1 Derangements We may use the principle of inclusion-exclusion to count the permutations of n objects that leave no objects in their original positions. Inserting these quantities into the formula for N(P ′1P ′ 2P ′ 3) shows that the number of solutions with x1 ≤ 3, x2 ≤ 4, and x3 ≤ 6 equals N(P ′1P ′ 2P ′ 3) = 78− 36− 28− 15 + 6 + 1 + 0− 0 = 6. N(P1P2P3) = number of solutions with x1 ≥ 4, x2 ≥ 5 and x3 ≥ 7 = 0.N(P2P3) = number of solutions with x2 ≥ 5 and x3 ≥ 7 = 0,.Using the same techniques as in Section 6.5, it follows that P ′ ik ) = N− ∑ 1≤i≤n N(Pi)+ ∑ 1≤i 3, property P2 if x2 > 4, and property P3 if x3 > 6. ![]() From the inclusion-exclusion principle, we see that N(P ′i1P ′ i2. Sup- pose the number of elements in the set is N. ![]() Let’s denote the number of elements with none of the properties Pi1, Pi2. Writing these quantities in terms of sets, we have |Ai1 ∩Ai2 ∩ Let’s denote the number of elements with all the properties Pi1, Pi2. Let Ai be the subset containing the elements that have property Pi. In particular, this form can be used to solve problems that ask for the number of elements in a set that have none of n properties P1, P2. An Alternative Form of Inclusion-Exclusion There is an alternative form of the principle of inclusion-exclusion that is useful in counting problems. This problem asks for the probability that no person is given the correct hat back by a hat-check person who gives the hats back randomly. The famous hat-check problem can be solved using the principle of inclusion-exclusion. Download Inclusion and exclusion principle and more Lecture notes Discrete Mathematics in PDF only on Docsity! 8.6 Applications of Inclusion-Exclusion Many counting problems can be solved using the principle of inclusion-exclusion.
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