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

Explanation:

Step1: List all possible combinations

A die has 6 faces numbered from 1 - 6. There are 4 cards numbered 4, 5, 6, 7.

Step2: Calculate the sums

For the card numbered 4: sums are 1 + 4=5, 2 + 4 = 6, 3+4 = 7, 4 + 4=8, 5 + 4=9, 6+4 = 10. For the card numbered 5: sums are 1 + 5=6 (already counted above), 2 + 5 = 7 (already counted above), 3+5 = 8 (already counted above), 4 + 5=9 (already counted above), 5 + 5=10 (already counted above), 6+5 = 11. For the card numbered 6: sums are 1 + 6=7 (already counted above), 2 + 6 = 8 (already counted above), 3+6 = 9 (already counted above), 4 + 6=10 (already counted above), 5 + 6=11 (already counted above), 6+6 = 12. For the card numbered 7: sums are 1 + 7=8 (already counted above), 2 + 7 = 9 (already counted above), 3+7 = 10 (already counted above), 4 + 7=11 (already counted above), 5 + 7=12 (already counted above), 6+7 = 13.

Step3: Count unique sums

The unique sums are 5, 6, 7, 8, 9, 10, 11, 12, 13. There are 9 unique sums.

Answer:

9