How many of the whole numbers from 1 to 300 are divisible by none of 2, 3 and 5?

Count the ones that ARE divisible by at least one of them, using inclusion–exclusion. Multiples of 2, 3 and 5 number 150, 100 and 60. Subtract those counted twice — multiples of 6, 10 and 15, which number 50, 30 and 20. Add back the multiples of 30, which number 10 and were removed once too often. That gives 150 + 100 + 60 − 50 − 30 − 20 + 10 = 220, so 300 − 220 = 80 numbers survive. A quick check: $300\times 1/2\times 2/3\times 4/5=80$. Reporting 220 answers the opposite question, and skipping the add-back step gives 60.