you pick one card from each set, roll a die, and find the sum. how many different sums are possible?

Explanation:

Step1: Find the minimum sum

The minimum - value card from the first set is 1, the minimum - value card from the second set is 1, and the minimum value on a die is 1. So the minimum sum is $1 + 1+1=3$.

Step2: Find the maximum sum

The maximum - value card from the first set is 4, the maximum - value card from the second set is 2, and the maximum value on a die is 6. So the maximum sum is $4 + 2+6 = 12$.

Step3: Analyze possible sums

The possible sums range from 3 to 12. We can check for non - existence of any values in this range. The set of all possible sums of three non - negative integers (from the two card sets and the die) will cover all integers in the range $[3,12]$. The number of integers from 3 to 12 is $12−3 + 1=10$.

Answer:

10