# What is the cardinality of given set

$S_n=\{x_i^{\alpha}x_j^{\gamma} \mid 0\leq \alpha , \gamma\leq n ~\& ~\alpha+\gamma=n ~\& ~ i,j \in \{1,2,3,4\} ~ \& ~ i \neq j\}$

For example $n=2$, $S_2=\{x_1^2,x_2^2,x_3^2,x_4^2,x_1x_2,x_1x_3,x_1x_4,x_2x_3,x_2x_4,x_3x_4\}$

What is the cardinality of $S_n?$

• Wait what is $x_i$? shouldn't it just be $i$? – Jorge Fernández Hidalgo Dec 18 '15 at 23:52
• Could you be more explicit? I don't understand exactly what you want. – Jorge Fernández Hidalgo Dec 18 '15 at 23:54

The definition of $S_n$ can be simplified a little: $$S_n=\{x_i^{\alpha}x_j^{n-\alpha} \mid 0\leq \alpha\leq n ~\& ~ i,j \in \{1,2,3,4\} ~ \& ~ i \neq j\}.$$
The number of $\alpha$ with $0\le\alpha\le n$ is $n+1$. For any finite set $K$, of size $k$, the number of $i,j\in K$ with $i\ne j$ is $k^2-k = k(k-1)$. Here, $K = \{1,2,3,4\}$, $k=4$.
So the size of $S_n$ is $12(n+1)$.