Rule of Sum and Rule of Product

Rule of Sum Problem Solving


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

How many triples of positive integers (a,b,c) (a, b, c) are there such that a2+b2c<4,a5,b5,c5?\begin{array}{c}&\frac{a^2 + b^2}{c}<4, &a\le 5, &b\le 5, &c\le 5? \end{array}

(A)   19\ \ 19
(B)   24\ \ 24
(C)   29\ \ 29
(D)   34\ \ 34
(E)   39\ \ 39

Famous musicians Kay-Zed and Beeta teamed up to do a concert together. During the concert, Kay-Zed played 1111 of his 4545 greatest hits, and Beeta played 1212 of her 5151 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 1515 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, BCABCA is a derangement of ABCABC.

Find the number of all the derangements of ABCD ABCD .

A number is picked randomly from 11 to 9,9, and then another number is picked from 11 to 9,9, 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?28?


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