\begin{align*} 1 = P(S) & = P( \{ x_1 \} \cup \{x_2 \} \cup ... \cup \{x_{|S|} \} )\\ & = P(\{ x_1 \}) + P(\{ x_2 \} ) + ... + P(\{ x_{|S|} \} )\\ & = c + c + ... + c = {|S|} \times c \end{align*}