Probability
# Rule of Sum and Rule of Product

How many triples of positive integers $(a, b, c)$ are there such that $\begin{array}{c}&\frac{a^2 + b^2}{c}<4, &a\le 5, &b\le 5, &c\le 5? \end{array}$

(A) $\ \ 19$

(B) $\ \ 24$

(C) $\ \ 29$

(D) $\ \ 34$

(E) $\ \ 39$

Famous musicians Kay-Zed and Beeta teamed up to do a concert together. During the concert, Kay-Zed played $11$ of his $45$ greatest hits, and Beeta played $12$ of her $51$ greatest hits. For an Encore performance, they are going to come out and perform a song together.

They will either perform one of Kay-Zed's greatest hits that he did not already perform, or one of Beeta's greatest hits that she did not already perform, or one from a list of $15$ other songs that the fans requested. How many choices for which song to perform do Kay-Zed and Beeta have?

A **derangement** of a string of distinct elements is a rearrangement of the string such that no element appears in its original position. For example, $BCA$ is a derangement of $ABC$.

Find the number of all the derangements of $ABCD$.