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

It is given that fourteen boys collected a total of 80 berries and each boy collected at least 1 berry.

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

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

If all 14 boys collected distinct number of berries, then the 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 pairs of boys collecting the same number of berries.

If 1 + 2 + 3 + 4 + ........ + 13 + 14 has to be reduced accordingly, then the highest distinct number of berries should be removed and the lowest 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 distinct number such that the sum is minimized:

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

i.e., 2 students collect a berry each, another set of 2 students collect 2 berries each, and the rest of the 10 students collect 3, 4, ....., 12 berries respectively.

Then, the total number of berries would be 92-13+2=81 (which is 1 greater than 80!!)

We want our total to be 80. We still need to bring down the total by 1.

In order to bring down the total by 1, the only way out is to remove 12 and add 11.

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

i.e., 2 students collect a berry each, another set of 2 students collect 2 berries each, another set of 2 students collect 11 berries each and the rest of the 8 students collect 3, 4, ....., 10 berries respectively.
Now, the total number of berries would be 81-12+11= 80. (Voilà!)

Thus, the minimum number of boys collecting the same number of berries will be 6 i.e., 3 pair of students will be collecting 1, 2 & 11 berries respectively.

The answer is Choice (D)