The problem involves distributing balls into a series of boxes so that each box has 4 balls more than the next one. This creates an arithmetic series. The sum of an arithmetic series can be given by the formula: n/2(2a+(n-1)d), where `n` is the number of terms (boxes), `a` is the first term, and `d` is the common difference.

Given we have 240 balls, the number of balls in the first box is 4(n-1) since each box has 4 balls more than the next one. Applying the sum formula, we get

240 = n/2 [2*4(n-1) + (n-1)4]

240 = 2n^2 - 2n

Solving this quadratic equation, we find no integer solutions for n=7. The answer options `4`, `5`, `6` yield integer solutions satisfying the condition, but `7` doesn`t. Hence, the number of boxes cannot be 7 i.e., option-4.