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# Fourteen boys went to collect berries and returned with a total of 80

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Re: Fourteen boys went to collect berries and returned with a total of 80 [#permalink]
Regor60 wrote:
RamnathWizako wrote:
From the given question, we have: Fourteen boys collected a total of 80 berries.
Each collected at least 1 berry.

Objective: To find a minimum number of pairs of boys that collected the same number of berries.

If the number of boys collecting a distinct number of berries is maximum then the number of boys collecting the same number of berries will be minimum.

If all 14 boys collected a distinct number of berries, then minimum number of berries collected by them will be
1 + 2 + 3 + 4 + ........ + 13 + 14 = $$\frac{(14∗15)}{2}$$ = 105

But, we have a total of only 80 berries. We have to bring down 105 to 80 with the least number of people collecting the same number of berries.

If 1 + 2 + 3 + 4 + ........ + 13 + 14 has to be reduced accordingly, then the maximum distinct number of berries should be removed and the minimum distinct number of berries collected should be duplicated.

1 2 3 4 5 6 7 8 9 10 11 12 13
1

i.e., 2 students collect a berry each and the rest of the 12 students collect 2, 3, 4, ....., 13 berries respectively.
Then, the total number of berries would be 105-14+1= 92. (which is still greater than 80)

Repeating the same operation once again, i.e., removing the greatest distinct number and adding the least similar number such that the sum is minimized:

1 2 3 4 5 6 7 8 9 10 11 12
1
1

i.e., 3 students collect a berry each and the rest of the 11 students collect 2, 3, 4, ....., 12 berries respectively.
Then, the total number of berries would be 92-13+1=80. (Voilà!)

Thus, the minimum number of boys collecting the same number of berries will be 3.

The question is asking for the minimum number of PAIRS of boys.

So choice D would suggest 3 pairs = 6 boys.

This question seems to be worded improperly.

Posted from my mobile device

Hello Regor60,
Thanks for pointing it out.
There is no error in the question.

Cheers.
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Re: Fourteen boys went to collect berries and returned with a total of 80 [#permalink]
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Each boy has at least one berry, so if we allocate each boy 1 berry after which we have 66 remaining
To have each boy have a unique number of berries with minimum berries we start allocating additional berries with the increase of one i.e. 1,2,3,4,5....... and on reaching 11, the total number of berries is exhausted, and we are left with 3 boys having 1 berry each which is option D
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Re: Fourteen boys went to collect berries and returned with a total of 80 [#permalink]
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Re: Fourteen boys went to collect berries and returned with a total of 80 [#permalink]
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