you roll a die, spin the spinner, pick a card, and find the sum. how many different sums are possible?

Explanation:

Step1: Determine number of outcomes for each event

Rolling a die has 6 outcomes, spinning the spinner has 2 outcomes (5 or 6), and picking a card has 2 outcomes (7 or 8).

Step2: Use the counting - principle

The total number of combined outcomes is the product of the number of outcomes of each event. So, the total number of ways to perform the three actions is $6\times2\times2=24$.

Step3: List out and find distinct sums

Let the die result be $d$ ($d = 1,2,\cdots,6$), spinner result be $s$ ($s=5$ or $s = 6$) and card result be $c$ ($c = 7$ or $c=8$). The sum $S=d + s + c$. When $s = 5,c = 7$: $S=d + 5+7=d + 12$, and $S$ ranges from $1+12 = 13$ to $6+12=18$. When $s = 5,c = 8$: $S=d + 5 + 8=d+13$, and $S$ ranges from $1+13 = 14$ to $6+13 = 19$. When $s = 6,c = 7$: $S=d + 6+7=d + 13$ (some overlap with previous case), and $S$ ranges from $1+13 = 14$ to $6+13 = 19$. When $s = 6,c = 8$: $S=d + 6+8=d + 14$, and $S$ ranges from $1+14 = 15$ to $6+14 = 20$. The distinct sums are 13, 14, 15, 16, 17, 18, 19, 20.

Answer:

8