Question 9

Formula:
1.True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in at least 1/2) - (# in at least 1/3) - (# in at least 2/3) + (# in 1/2/3)

2.True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) - 2(# in 1/2/3)

examples:

At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry and English. 53 students study math, 88 study chemistry and 58 study english.

If 6 students take all 3 classes, how many take exactly 2 classes?

In this case, we'd use the first formula, since we want the number who take exactly 2 classes:  
150 = 53 + 88 + 58 - (doubles) - 2(triples) 
150 = 199 - (doubles) - 2(6) 
150 = 187 - doubles  doubles = 37  
Let's just change the question a tiny bit:

At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry and English. 53 students study math, 88 study chemistry and 58 study english.

If 6 students take all 3 classes, how many take at least 2 classes?

In this case, we'd use the second formula, since we want the number who take at least 2 classes:  
150 = 53 + 88 + 58 - (at least 2 of the 3) + (all 3) 
150 = 199 - (at least 2 of 3) + 6 
150 = 193 - (at least 2 of 3) 
At least 2 of 3 = 43