Final answer:
To determine the number of visually distinct permutations for Caleb's row of yellow and red balls, the factorial of the total number of balls is divided by the product of the factorials of the number of balls of each color, resulting in 35 distinct arrangements.
Step-by-step explanation:
Caleb has placed three yellow balls and four red balls to form a row. The question is how many visually distinct rows are possible with these balls. To find the solution, we need to determine the number of different permutations that can be created with the balls, considering that balls of the same color are indistinguishable from one another.
The total number of ways to arrange the seven balls if they were all distinct would be 7 factorial (7!). Since the balls include groups of the same colors, we should divide this number by the factorial of the occurrences of each color, which is 3! for the three yellow balls and 4! for the four red balls. This is an application of the permutation formula for indistinguishable objects.
Thus the formula to calculate the visually distinct rows is:
Distinct permutations = 7! / (3! × 4!) = 5,040 / (6 × 24) = 5,040 / 144 = 35.
Therefore, there are 35 visually distinct rows possible.