If W is the set of all the integers between 49 and 76, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?  Choose the correct answer.
A) 19 B) 91 C) 41 D) 18 E) 14

Rationale: "official solution : number of multiples of 3 step 1 . subtract the extreme multiples of 3 within the range ( the greatest is 75 , the smallest is 51 ) : 75 - 51 = 24 step 2 . divide by 3 : 24 / 3 = 8 step 3 . add 1 : 8 + 1 = 9 . so there are 9 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 76 , the smallest is 50 ) : 76 - 50 = 26 step 2 . divide by 2 : 26 / 2 = 13 step 3 . add 1 : 13 + 1 = 14 . so there are 14 multiples of 2 within the range : examples are 50 , 52 , 54 , 56 , 58 , 60 etc . add the 9 multiples of 3 and the 14 multiples of 2 : 9 + 14 = 23 . 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 9 + 14 = 23 . 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 23 : step 1 . subtract the extreme multiples of 6 within the range ( the greatest is 72 , the smallest is 54 ) : 72 - 54 = 18 step 2 . divide by 6 : 18 / 6 = 3 step 3 . add 1 : 3 + 1 = 4 . so there are 4 multiples of 6 within the range : we counted 4 numbers twice . subtract the 4 multiples of 6 from the sum of the multiples of 2 and 3 : = 9 + 14 - 4 = 23 - 4 = 19 therefore , the final number of multiples of 2 , 3 or 6 is 19 . hence , this is the correct answer . ( a )"
Answer: A