you roll two dice, spin the spinner, and find the sum. how many different sums are possible?

Answer

Explanation:

Step1: Find range of dice - sum

The minimum value when rolling two dice is (1 + 1=2), and the maximum value is (6 + 6 = 12). So the possible sums of two - dice rolls are in the set ({2,3,\cdots,12}).

Step2: Consider spinner values

The spinner has values (2,3,4,5,6,7).

Step3: Find minimum and maximum combined sums

The minimum combined sum is (2+2 = 4) (minimum dice - sum plus minimum spinner value). The maximum combined sum is (12 + 7=19) (maximum dice - sum plus maximum spinner value).

Step4: Analyze all possible sums

We will list out the sums. Let (d) be the sum of the two dice ((2\leq d\leq12)) and (s) be the value on the spinner ((2\leq s\leq7)). The combined sum (C=d + s). When (d = 2): (C) values are (2 + 2=4,2 + 3 = 5,2+4 = 6,2 + 5=7,2 + 6 = 8,2+7 = 9). When (d = 3): (C) values are (3 + 2=5,3 + 3 = 6,3+4 = 7,3 + 5=8,3 + 6 = 9,3+7 = 10). We can observe that we can get all integer values from (4) to (19). The number of integers from (4) to (19) is (19 - 4+1=16).