Final answer:
The total number of ways a customer can choose 4 ice-cream cones with only 2 or 3 different flavors out of 6 flavors is 2460 ways.
Step-by-step explanation:
To find the number of ways a customer can choose 4 ice-cream cones with only 2 or 3 different flavors out of 6 flavors, we need to consider two cases: when there are 2 different flavors and when there are 3 different flavors.
Case 1: 2 different flavors - There are 6C2 = 15 ways to choose 2 flavors out of 6. For each flavor, there are 4 cones to choose from. So, there are 4C2 = 6 ways to choose 2 cones of the first flavor and 4C2 = 6 ways to choose 2 cones of the second flavor. Therefore, the total number of ways is 15 * 6 * 6 = 540 ways.
Case 2: 3 different flavors - There are 6C3 = 20 ways to choose 3 flavors out of 6. For each flavor, there are 4 cones to choose from. So, there are 4C1 = 4 ways to choose 1 cone of the first flavor, 4C1 = 4 ways to choose 1 cone of the second flavor, and 4C2 = 6 ways to choose 2 cones of the third flavor. Therefore, the total number of ways is 20 * 4 * 4 * 6 = 1920 ways.
The total number of ways to choose 4 ice-cream cones with 2 or 3 different flavors is 540 + 1920 = 2460 ways.