Q.A coin and a die are thrown together (simultaneously). Write the sample space of this experiment, and find $n(S)$.

[!TLDR] The coin gives $2$ outcomes and the die gives $6$; pair every coin outcome with every die outcome. [!ANSWER] $S= …

The coin shows H or T ($2$ outcomes), and independently the die shows one of $1,\dots,6$ ($6$ outcomes). By the multiplication principle, the total number of outcomes is $2 \times 6 = 12$. Listing every coin-die combination: $S = {(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),( …

Use the multiplication principle: pair every outcome of the first part of the experiment (coin) with every outcome of …

A common mistake is listing only $6$ or $8$ outcomes by not pairing every coin face with every die face — go through H-with-each-die-face …