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# Rule of Sum and Rule of Product

Seek simple and succinct solutions in these systems by sussing-out the significant symmetries.

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

(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 \( ABCD \).

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