TIFR 2011

Question

There are N boxes, each containing at most r balls. If the number of boxes containing at least i balls is $N_{i}$ for i = 1,2,...r, then the total number of balls contained in these N boxes

(A) is exactly equal to $N_{1}$ +$N_{2}$ +...+$N_{r}$.

(B) is strictly larger than $N_{1}$ +$N_{2}$ +...+$N_{r}$.

(C) is strictly smaller than $N_{1}$ +$N_{2}$ +...+$N_{r}$.

(D) cannot be determined from the given information

Answer 1

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 $N_{1} + N_{2} + N_{3} + … + N_{r}$