Answer: Option (A)
Given $N_{i}$ is the number of boxes that contain atleast $i$ balls for $i=1,2,3…, r$
Let, number of boxes that contain exactly 1 ball = $x_{1}$
number of boxes that contain exactly 2 balls = $x_{2}$ and so on.
Total number of boxes = $x_{1} + 2*x_{2} + 3*x_{3} + … + r*x_{r}$
Number of boxes that contain atleast 1 ball = number of boxes with exactly 1 ball + number of boxes with exactly 2 balls + …
Therefore,
$x_{1} + x_{2} + x_{3} + … + x_{r} = N_{1}$
$x_{2} + x_{3} + x_{4} + … + x_{r} = N_{2}$
.
.
.
$n_{r} = N_{r}$
Adding above eqations we get
$x_{1} + 2*x_{2} + 3*x_{3} + … + r*x_{r} = N_{1} + N_{2} + N_{3} + … + N_{r}$
Therefore total number of balls are exactly $N_{1} + N_{2} + N_{3} + … + N_{r}$