Be careful

Probability Level 4

There are $N$ boxes, each containing at most $r$ balls. The number of boxes containing at least $i$ balls, denoted as $N_i,$ is exactly $i$ for $i=1,2,\ldots,r$ .

The total number of balls contained in these $N$ boxes:

is strictly smaller than

N_1+N_2+\ldots+N_r

cannot be determined from the given information is strictly greater than

N_1+N_2+\ldots+N_r

is exactly equal to

N_1+N_2+\ldots+N_r

1 solution

Kalpok Guha
Oct 8, 2015

Number of boxes containing exactly $1$ ball is $N_1-N_2$

Number of boxes containing exactly $2$ balls is $N_2-N_3$

Number of boxes containing exactly $r$ balls is $N_{r-1}-N_r$

Hence the total number of balls in $r$ boxes is $(N_1-N_2)+2(N_2-N_3)+\ldots+r(N_{r-1}-N_r)=N_1+N_2+\ldots+N_r$

The first solution by Kalpok Guha belongs to a slightly different question, as the current question contains a typo. The correct version of the question should start like "Number of boxes containing at most i balls", instead of " at least i balls".

"Number of boxes containing at least i balls, denoted as Ni is exactly i for i=1,2,3,...,r" means, that the number of boxes containing at least 1 ball is 1, the number of boxes containing at least 2 balls is 2 ... and so on

Here we have an impossible situation, as those boxes, which contain at least 2 balls also contain at least 1 ball. Therefore, N1 cannot be 1 (by the way, if Nr=r, then Ni=r for i=1,2,3,...,r).

Hence, the closest thing to the correct answer for the current wording is that it "Cannot be determined from the given information".

Zee Ell - 5 years, 7 months ago

Be careful

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