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

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Manager
Joined: 18 Jul 2019
Posts: 54
Fourteen boys went to collect berries and returned with a total of 80  [#permalink]

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25 Nov 2019, 06:13
5
00:00

Difficulty:

65% (hard)

Question Stats:

32% (01:37) correct 68% (01:43) wrong based on 25 sessions

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Fourteen boys went to collect berries and returned with a total of 80 berries among themselves. If every boy collected at least one berry each, then what is the minimum number of pairs of boys that must have collected the same number of berries?

a) 0
b) 1
c) 2
d) 3
e) 4
Math Expert
Joined: 02 Aug 2009
Posts: 8305
Re: Fourteen boys went to collect berries and returned with a total of 80  [#permalink]

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26 Nov 2019, 18:41
CaptainLevi wrote:
Fourteen boys went to collect berries and returned with a total of 80 berries among themselves. If every boy collected at least one berry each, then what is the minimum number of pairs of boys that must have collected the same number of berries?

a) 0
b) 1
c) 2
d) 3
e) 4

Let each have minimum number possible and different from each other, then total = 1+2+3+4..+14=14*15/2=105
In this case we required 105-80 or 25 more

So we can give the ones with maximum number the least possible..
14 be given 1, so 105-14+1=92...12 extra
13 be given 1, so 92-13+1=80...exact
Here 3 boys have 1 each, and hence 3C2 or 3 pairs with same number of berries
14 be given 1, so 105-14+1=92...12 extra
13 be given 2, so 92-13+2=81...1 extra
12 be given 11, so 81-12+11=80..Exact
Here 3 pair of boys have same number of berries - 1 each, 2each and 11 each

D
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Re: Fourteen boys went to collect berries and returned with a total of 80   [#permalink] 26 Nov 2019, 18:41
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