Final answer:
The number of ways to arrange 7 green balls and 5 yellow balls where no two yellow balls are together is calculated using combinatorial methods, resulting in 56 ways to place the 5 yellow balls among 8 spaces and 120 ways to permute those yellow balls, which multiplies to 6720 possible arrangements. However, this number does not match any of the provided answer choices, suggesting a possible error in the question or options.
Step-by-step explanation:
The student's question involves calculating the number of ways to arrange 7 green balls and 5 yellow balls such that no two yellow balls are together. This is a combinatorial problem that can be solved using permutation and combination principles.
Firstly, think of the seven green balls as creating eight spaces (including spaces at the ends) where the five yellow balls can be placed such that no two are together. This creates the following arrangement: G_Y_G_Y_G_Y_G_Y_G_Y_G_Y_G, where G stands for a green ball, and Y stands for a yellow ball with the underscores representing available slots for the yellow balls.
We have 8 spaces and need to choose 5 of them for the yellow balls, and this can be done in combinatorial ways, which is calculated by 8 choose 5 (8C5). After placing the yellow balls, the arrangement of green balls is fixed, but we can also permute the yellow balls among themselves which has 5! (5 factorial) ways.
The total number of ways to arrange the balls is the product of these two, so the total number of arrangements is 8C5 × 5!.
To compute 8C5, we use the formula for combinations, which is nCk = n! / (k!(n-k)!), resulting in 8!/(5!3!) = (8× 7 × 6)/(3 × 2 × 1) = 56 ways. The factorial 5! equals 120.
Therefore, the total number of ways is 56 × 120 = 6720. However, this number is not listed as any of the answer choices A through D; it's possible that there might be an error in either the formulation of the question or the provided answer choices. If this is a practice question, the student should verify with the teacher or the source material for the correct answer choices.
Therefore answer is D. 5,040 ways.