A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples.
My answer is:
First distribute $90$ apples so that each will have $6$ apples. Now we have $100-90 = 10$ apples. Now we are distributing these many apples between two each will get $5$ apples so the ans is $2$ boys will get $11$ apples each and other $13$ boys will get $6$ apples each... Is this Proof correct ????
Your argument is incorrect.
Suppose that each boy plucked a different number of apples. Then the smallest total the group of boys could have plucked is $$0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 105$$ However, they collectively plucked just $100$ apples. Hence, two boys must have plucked the same number of apples.