 Probability

# 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 $\begin{array}{c}&\frac{a^2 + b^2}{c}<4, &a\le 5, &b\le 5, &c\le 5? \end{array}$

(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 $ABCD$.

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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