Let X be a set with n elements, n > 3. Determine the size of the following set in terms of n, and give a big-Θ estimate for the answer. the set of all strings x1x2, with xi is in X Θ(n!) Θ(log2 n) Θ(n3) Θ(n2) Θ(2n)

Question
Sequences
asked 2021-01-10
Let X be a set with n elements, n > 3. Determine the size of the following set in terms of n, and give a \(\displaystyle{b}{i}{g}-Θ\) estimate for the answer. the set of all strings x1x2, with xi is in \(\displaystyle{X}Θ{\left({n}!\right)}Θ{\left({\log{{2}}}{n}\right)}Θ{\left({n}{3}\right)}Θ{\left({n}{2}\right)}Θ{\left({2}{n}\right)}\)

Answers (1)

2021-01-11
Fundamental counting principle: If the first event could occur in m ways and the second event could occur in n ways, then the number of ways that the two events could occur in sequence is m*n.
Solution
X is a set with n elements (n>3).
We are interested in the number of elements in the set \(\displaystyle{\left\lbrace{x}{1}{x}{2}{\mid}{x}{1},{x}{2}∈{X}\right\rbrace}\). Since X contains n elements, there are n options for x1 and n options for x2.
x1: n ways
x2: n ways
By the fundamental counting principle:
\(\displaystyle{n}\cdot{n}={n}^{{2}}\)
Thus there are \(\displaystyle{n}^{{2}}\) strings in the set \(\displaystyle{\left\lbrace{x}{1}{x}{2}{\mid}{x}{1},{x}{2}∈{X}\right\rbrace}\), while \(\displaystyle{n}^{{2}}\) is \(\displaystyleƟ{\left({n}^{{2}}\right)}\)
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