If W is the set of all the integers between 10 and 332, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?
Select
A) 111	B) 28	C) 160	D) 213	E) 107

Rationale: "official solution : number of multiples of 3 step 1 . subtract the extreme multiples of 3 within the range ( the greatest is 330 , the smallest is 12 ) : 330 - 12 = 318 step 2 . divide by 3 : 318 / 3 = 106 step 3 . add 1 : 106 + 1 = 107 . so there are 107 multiples of 3 within the range : examples are 51 , 54 , 57 , 60 , etc . number of multiples of 2 step 1 . subtract the extreme multiples of 2 within the range ( the greatest is 330 , the smallest is 12 ) : 330 - 12 = 318 step 2 . divide by 2 : 318 / 2 = 159 step 3 . add 1 : 159 + 1 = 160 . so there are 160 multiples of 2 within the range : examples are 50 , 52 , 54 , 56 , 58 , 60 etc . add the 107 multiples of 3 and the 160 multiples of 2 : 107 + 160 = 267 . however , by adding the multiples of 2 and the multiples of 3 , we are effectively counting several numbers twice : for example , 54 and 60 are parts of both the lists above . so we ca n ' t just take 107 + 160 = 267 . find the number of multiples of 6 ( which are the double counted , as 6 is divisible by both 2 and 3 ) , and subtract it from 25 : step 1 . subtract the extreme multiples of 6 within the range ( the greatest is 72 , the smallest is 54 ) : 330 - 12 = 318 step 2 . divide by 6 : 318 / 6 = 53 step 3 . add 1 : 53 + 1 = 54 . so there are 54 multiples of 6 within the range : we counted 54 numbers twice . subtract the 54 multiples of 6 from the sum of the multiples of 2 and 3 : = 107 + 160 - 54 = 267 - 54 = 213 therefore , the final number of multiples of 2 , 3 or 6 is 213 . hence , this is the correct answer . ( d )"
Answer: D