Question

Let $S$ be a finite set containing $n$ elements. Then, the total number of commutative binary operation on $S$ is:

• A. $n^3$
• B. $n^2$
• C. $n^{n^2}$
• D. $n^n$

Answer 1

The correct option is C

$n^{n^2}$

Explanation For The Correct Option:

Finding the total number of commutative binary operation on $S$

Given that cardinality of $\left|S\right|$$=n$

Since commutative binary operation

$\Rightarrow S\times S\to S$

Therefore,

$$\begin{gathered} \Rightarrow {\left|s\right|}^{\left|S\times S\right|} \\ \Rightarrow n^{n\times n}\left[\because \left|s\right|=n\right] \\ \Rightarrow n^{n^2} \end{gathered}$$

Hence, option C is the correct answer.