Find the number of positive integers n between $1$ and $100$ that are not a multiple of $3$ or $11$.
My Try:
Numbers divisible $3$ between $1$ and $100=33$ (set A)
Numbers divisible $11$ between $1$ and $100=9$ (set B)
Numbers divisible $3$ and $11$ between $1$ and $100=2$ (set A and B)
A U B = A + B + A and B
A U B $= 33 + 9 + 3 = 45$
The numbers that are not divisible by $3$ and $11$ between $1$ and $100$ are $100-45=55$.
But the given answer is $61$
Can anyone show where I was wrong?
I'm assuming that there was a typo, and you meant to put $|A \cap B| = 3$ and not $2$, in both places.
Your main problem is that you've applied the principle of inclusion and exclusion incorrectly, and
$$ |A \cup B| = |A| + |B| - |A \cap B| = 33 + 9 - 3 = 39 $$
and not $33 + 9 + 3 = 45$. After that, you should obtain the correct answer.