Question

Let $\text{S}$ be a finite set containing $\text{n}$ elements. Then, the total number of commutative binary operation on $\text{S}$ 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 correct answer Option:

Find the total number of commutative binary operation on $\text{S}$.

Number of binary operations can be expressed as,

$$\begin{gathered} \Rightarrow S\times S\to S \\ \Rightarrow n{\left(S\right)}^{\left[n\left(S\times S\right)\right]} \\ \Rightarrow n^{\left[n\left(S\right)\times n\left(S\right)\right]} \\ \Rightarrow n^{n\times n} \\ \Rightarrow n^{n^2} \end{gathered}$$

Therefore, the correct answer is Option C.