$|P(A)| = 2^{|A|}$ question For the following question:
Let $A = \{ 1,2,3,4,5 \}$, $B = \{ a,b,c,d,e \}$, and $C = \{ \text{red} , \text{green} , \text{blue} \}$.  Calculate $| \mathcal{P} ( A \times B \times C ) |$.
I know $|P(A)| = 2^{|A|}$
So am I right in thinking ... $5\times5\times3 = 75 = 2^{75}$ would be my answer?

You have the right idea. The only problem is saying that $5\times 5\times 3=75=2^{75}$. Instead, notice that $|A|=5$, $|B|=5$, and $|C|=3$. We know that $|A\times B\times C|=75$ by the definition of Cartesian Product. Thus $|\mathcal{P}(A\times B\times C)|=2^{75}$.
