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

Answer

Explanation:

Step1: Determine the number of outcomes for each action

There are 2 cards (3 and 4), so 2 outcomes for picking a card. There are 2 sections on the spinner (7 and 8), so 2 outcomes for spinning the spinner. There are 6 sides on a die, so 6 outcomes for rolling a die.

Step2: Calculate the total number of combinations

By the fundamental counting principle, the total number of combinations of picking a card, spinning the spinner and rolling a die is $2\times2\times6=24$.

Step3: Find the minimum and maximum sums

The minimum sum occurs when we pick 3 (card), spin 7 (spinner) and roll 1 (die): $3 + 7+1=11$. The maximum sum occurs when we pick 4 (card), spin 8 (spinner) and roll 6 (die): $4 + 8+6=18$.

Step4: List out possible sums and count unique values

The possible sums range from 11 to 18. The possible sums are:

When card = 3, spinner = 7:
- die = 1, sum = $3 + 7+1 = 11$; die = 2, sum = $3 + 7+2 = 12$; die = 3, sum = $3 + 7+3 = 13$; die = 4, sum = $3 + 7+4 = 14$; die = 5, sum = $3 + 7+5 = 15$; die = 6, sum = $3 + 7+6 = 16$.
When card = 3, spinner = 8:
- die = 1, sum = $3 + 8+1 = 12$; die = 2, sum = $3 + 8+2 = 13$; die = 3, sum = $3 + 8+3 = 14$; die = 4, sum = $3 + 8+4 = 15$; die = 5, sum = $3 + 8+5 = 16$; die = 6, sum = $3 + 8+6 = 17$.
When card = 4, spinner = 7:
- die = 1, sum = $4 + 7+1 = 12$; die = 2, sum = $4 + 7+2 = 13$; die = 3, sum = $4 + 7+3 = 14$; die = 4, sum = $4 + 7+4 = 15$; die = 5, sum = $4 + 7+5 = 16$; die = 6, sum = $4 + 7+6 = 17$.
When card = 4, spinner = 8:
- die = 1, sum = $4 + 8+1 = 13$; die = 2, sum = $4 + 8+2 = 14$; die = 3, sum = $4 + 8+3 = 15$; die = 4, sum = $4 + 8+4 = 16$; die = 5, sum = $4 + 8+5 = 17$; die = 6, sum = $4 + 8+6 = 18$. The unique sums are 11, 12, 13, 14, 15, 16, 17, 18. So there are 8 different sums.