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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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