Question 1180837: Of the 30 students in my class, 11 practice taekwondo, 11 play piano, and 20 take lessons at AoPS. 4 students play piano and go to AoPS, 3 students do taekwondo and play piano, 5 student go to AoPS and practice taekwondo, and 3 students do none of these. How many students do all 3?
Answer by ikleyn(52878)   (Show Source):
You can put this solution on YOUR website! .
Of the 30 students in my class, 11 practice taekwondo, 11 play piano, and 20 take lessons at AoPS.
4 students play piano and go to AoPS, 3 students do taekwondo and play piano, 5 student go to AoPS and practice taekwondo,
and 3 students do none of these. How many students do all 3?
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In this problem, you are given the number of elements of the entire "universal" set (class);
the number of elements in its three subsets (11, 11 and 20);
the number of elements in in-pairs intersections of these three subsets (4, 3 and 5),
the number of elements in the UNION of the three subsets (27 = 30-3).
They want you find the number of elements " x " in the triple intersection of subsets.
Apply the inclusion-exclusion principle
27 = 11 + 11 + 20 - 4 - 3 - 5 + x (alternate sum)
or
27 = 30 + x
This equation has no solution in integer non-negative numbers.
THEREFORE, the posed problem HAS NO SOLUTIONS.
It describes a situation, which NEVER MAY HAPPEN.
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On Inclusion-Exclusion principle, read this Wikipedia article
https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
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Had your problem be posed correctly, I would be happy to solve it and to teach you.
Moreover, had it be posed correctly, I would add the list of relevant lessons
and similar solved problems from the archive of this forum as a bonus for you.
But at the given badly circumstances, I will not do it . . .
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