# Am I applying the Inclusion Exclusion principle correctly?

Q: Find how many integers $n$ satisfying $1\le n \le 5000$ are divisible by at least one of the numbers 4, 7 and 33.

I've done the following:

$$|A| = \lfloor 5000/4 \rfloor = 1,250$$ $$|B| = \lfloor 5000/7 \rfloor = 714$$ $$|C| = \lfloor 5000/33 \rfloor = 151$$

$$|A\cap B| = \lfloor 5000/(4 * 7) \rfloor = 178$$ $$|A\cap C| = \lfloor 5000/(4 * 33) \rfloor = 37$$ $$|B\cap C| = \lfloor 5000/(7 * 33) \rfloor = 21$$ $$|A \cap B\cap C| = \lfloor 5000/(4 * 7 * 33) \rfloor = 5$$

Next I do this:

$$|A|+|B|+|C| - |A\cap B| - |A\cap C| - |B\cap C| + |A \cap B\cap C|$$

Which works out as:

$$1,250 + 714 + 151 - 178 - 37 - 21 + 5 = 1,884$$