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

Answer

Explanation:

Step1: Find total number of combinations

There are 3 values on the first - spinner, 6 values on the second - spinner, and 7 cards. The total number of combinations of spinning the spinners and picking a card is (3\times6\times7 = 126) combinations. But we are interested in the sums. Let the values on the first spinner be (a\in{6,7,8}), the values on the second spinner be (b\in{3,4,5,6,7,8}), and the values on the cards be (c\in{2,3,4,5,6,7,8}). The sum (S=a + b + c).

Step2: Find the minimum sum

The minimum value of (a) is 6, the minimum value of (b) is 3, and the minimum value of (c) is 2. So the minimum sum (S_{\min}=6 + 3+2=11).

Step3: Find the maximum sum

The maximum value of (a) is 8, the maximum value of (b) is 8, and the maximum value of (c) is 8. So the maximum sum (S_{\max}=8 + 8+8 = 24).

Step4: Check for all possible sums

We can list out all the possible sums by considering all combinations. The possible sums range from 11 to 24. We can check if all values in this range can be formed. For (S = 11): (a = 6,b = 3,c = 2) For (S = 12): (a = 6,b = 3,c = 3) etc. The number of integers from 11 to 24 is (24 - 11+1=14).