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

Explanation:

Step1: Determine number of outcomes for each action

Spinner has 2 outcomes (3 or 4), die has 6 outcomes (1 - 6), cards have 2 outcomes (6 or 7).

Step2: Calculate total number of combinations

By multiplication - principle, total combinations are $2\times6\times2=24$.

Step3: Find the minimum and maximum sums

Minimum sum: $3 + 1+6 = 10$. Maximum sum: $4 + 6+7 = 17$.

Step4: List out possible sums and count distinct ones

List out all 24 sums by considering all combinations. Sums range from 10 to 17. The possible sums are 10, 11, 12, 13, 14, 15, 16, 17.

Answer:

8