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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.

Rule of Sum Problem Solving

Let \(A\) be the set of consecutive integers \( \{16,17,\ldots,51\}\) and \(B \) be the set of consecutive integers \(\{70, 71, \ldots, 111\}.\) How many different integers are in \(A\) or \(B?\)

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 \( ABCD \).

A number is picked randomly from \(1\) to \(9,\) and then another number is picked from \(1\) to \(9,\) regardless of the first pick. If the first pick is the ten's digit of a two-digit number and the second pick is the units digit, how many cases are there such that the two-digit number is less than \(28?\)

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