Problem 22
The Hatcheck Problem
At a party with a very large number of guests, everyone has to leave their hat at the entrance. When they leave the party, everyone is too drunk to remember what hat is theirs, so they just pick one at random. What is the probability that no one picks their own hat?

1−xN+1 2 N 1�∞n 1−x =1+x+x +…+x 1−x= x
n=0 ∞∞
1 = �anxn 1 = �xrn 1−ax 1−xr
n=0� �
(1+x)r = ∞ r xn where m =m(m−1)···(m−k+1)
� � n=0 n=0n k k!
1 �∞�n+r−1�n √ �∞�2n�(−1)n+1 n (1−x)r = n x 1+x= n 4n(2n−1)x
n=0 n=0
e x = �∞ x n n=0 n!
�∞ x2n+1
= (2n + 1)! n=0
ex +e−x �∞ x2n cosh(x ) = 2 = (2n)!
n=0
sinh(x) =
ex −e−x 2

Problem 22
The Hatcheck Problem
At a party with a very large number of guests, everyone has to leave their hat at the entrance. When they leave the party, everyone is too drunk to remember what hat is theirs, so they just pick one at random. What is the probability that no one picks their own hat?

1−xN+1 2 N 1�∞n 1−x =1+x+x +…+x 1−x= x
n=0 ∞∞
1 = �anxn 1 = �xrn 1−ax 1−xr
n=0� �
(1+x)r = ∞ r xn where m =m(m−1)···(m−k+1)
� � n=0 n=0n k k!
1 �∞�n+r−1�n √ �∞�2n�(−1)n+1 n (1−x)r = n x 1+x= n 4n(2n−1)x
n=0 n=0
e x = �∞ x n n=0 n!
�∞ x2n+1
= (2n + 1)! n=0
ex +e−x �∞ x2n cosh(x ) = 2 = (2n)!
n=0
sinh(x) =
ex −e−x 2