You roll a die with 6 sides, spin the spinner with 4,5,6,7,8, pick a card with 8 cards, and find the sum. How many different sums are possible?
There are a few different ways to approach this problem, but one method is to use a table to list out all the possible outcomes and their sums.
First, list out the possible outcomes of rolling the die:
1, 2, 3, 4, 5, 6
Next, list out the possible outcomes of spinning the spinner:
4, 5, 6, 7, 8
Then, list out the possible outcomes of picking a card from the deck:
1, 2, 3, 4, 5, 6, 7, 8
Now we can create a table to list all the possible combinations and their sums:
| Die | Spinner | Card | Sum |
|-----|---------|------|-----|
| 1 | 4 | 1 | 6 |
| 1 | 4 | 2 | 7 |
| 1 | 4 | 3 | 8 |
| 1 | 4 | 4 | 9 |
| 1 | 4 | 5 | 10 |
| 1 | 4 | 6 | 11 |
| 1 | 4 | 7 | 12 |
| 1 | 4 | 8 | 13 |
| 1 | 5 | 1 | 7 |
| 1 | 5 | 2 | 8 |
| ... | ... | ... | ... |
Continuing to list out all possible combinations and their sums would be very time-consuming, so we can look for patterns and shortcuts.
One pattern we can see is that adding 1 to the die roll will always increase the sum by 3. For example, the sum of rolling a 1 with a spinner value of 4 and a card value of 1 is 6, and the sum of rolling a 2 with the same spinner and card values is 9 (6 + 3). We can use this pattern to fill out some of the missing values in the table:
| Die | Spinner | Card | Sum |
|-----|---------|------|-----|
| 1 | 4 | 1 | 6 |
| 1 | 4 | 2 | 7 |
| 1 | 4 | 3 | 8 |
| 1 | 4 | 4 | 9 |
| 1 | 4 | 5 | 10 |
| 1 | 4 | 6 | 11 |
| 1 | 4 | 7 | 12 |
| 1 | 4 | 8 | 13 |
| 2 | 4 | 1 | 7 |
| 2 | 4 | 2 | 8 |
| 2 | 4 | 3 | 9 |
| ... | ... | ... | ... |
Similarly, we can see that each increase of 1 in the spinner value will increase the sum by 1. And since there are 8 cards with values 1-8, each value will appear exactly once in the sum.
Using these patterns, we can quickly determine that the possible sums are:
6, 7, 8, 9, 10, 11, 12, 13, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 11, 12, 13, 14, 15, 16, 17, 18
So there are 43 different possible sums.