5th ed., Ch. 5 #8 page 93 - THREE Overlapping Sets formula?

[adt29]

The 38 movies in a video store fall into the following 3 categories: 10 action, 20 drama, and 18 comedy. however, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy and 4 are both drama and comedy. how many action-drama-comedies are there?

In my experience, the Venn diagram method always takes longer than two minutes and so I'd like to use a formula. Browsing the forums, I came across this link: overlapping-sets-t1960.html

which gives me two formulas for overlapping sets:
Formula 1:

# of items total = (a + b + c) - (ab + ac + bc) + (abc)

in the formula, 'a' stands for the # of items in set a (regardless of whether they are in other sets as well), 'ab' for the # of items that are in both sets a and b (again, regardless of whether they're in c), and so forth.

note that you cannot use this formula if you are given information with exclusivity (for instance, the number of items in set a and b but not in set c).

Formula 2:

Total = H + M + E - only (H intersect M) - only (H intersect E) - only (M intersect E) - 2 * (H intersect M intersect E)

only (H intersect M) implies
(H intersect M) - (H intersect M intersect E)

My question is: how do I know which formula to use? What language or wording can I pick up on in the question that will help me choose the right one? Thank you!

[tim]

I typically don't recommend using any formulas on this type of problem, but if you're going to use a formula, feel free to memorize both of these and use whichever one you want. They're interchangeable, and it may be that one feels better for a given problem, so just make your choice and run with it! :)

[adt29]

Thanks, Tim. I should have added this to my initial post: plugging in numbers in both formulas doesn't give me the same answer.

The correct answer is 2.

Using Formula 1, I get x=2, if action-drama-comedies are x.

# of items total = (a + b + c) - (ab + ac + bc) + (abc)
38 = (10+20+18) - (5+3+4) + x
38 = 48 - 12 + x
38 = 36 + x
x = 2

Using Formula 2, I get x as a nonsensical number.
Total = H + M + E - only (H intersect M) - only (H intersect E) - only (M intersect E) - 2 * (H intersect M intersect E)

38 = 10+20+18 -5-3-4-2*x
38 = 48-12-2*x
38 - 36 = -2*x
2 = -2x
x = -1

Obviously formula 1 gets me the right answer, but I'm not sure how or why. My guess is that using Formula 2, I would need to modify some part of the formula, but again I'm not sure how or why.

Can you tell me what I'm missing in terms of my understanding?

Thanks!

[RonPurewal]

the problem with your formula #2 here is that you are using 5, 3, and 4 as the values for "only intersect blah blah blah".
that's wrong; those are not exclusive.
in formula #2 -- as you wrote above yourself -- "only intersect this and that" refers only to things that are in this and that but that aren't in the third set.
the problem here is that there are 2 values that are in all three sets. so, if you were to use this formula here, the numbers you'd use would be 3, 1, and 2, rather than 5, 3, and 4.
... and of course you can't do that anyway, because you don't have the "2" until you solve the problem.

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[RonPurewal]

in any case, if you can successfully memorize these formulas, you're a better person than i am.
there's no way i could possibly memorize something that long and obscure.

... but the good news is that you don't have to.
you can derive either one of these formulas by just drawing the 3-set venn diagram and watching what happens to it.

for instance:

in formula 1, after you add a + b + c you've counted...
... the things that are in only 1 set exactly once,
... the things that are in 2 sets exactly twice, and
... the things that are in all 3 sets three times.

then, after you subtract ab, ac, and bc, you have...
... (still) the things that are in only 1 set exactly once,
... the things that are in 2 sets exactly once (because each has just been subtracted once), and
... the things that are in all 3 sets, not at all; you subtracted all 3 instances back out.

so, all that's left is to add abc back in.
that's where that formula comes from.

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[RonPurewal]

finally, this:

In my experience, the Venn diagram method always takes longer than two minutes

... you say that as though it were a bad thing.

remember, "2 minutes" is an average time for quant problems.
that's average as in average. not maximum!

trying to solve every problem in 2 minutes or less is a VERY bad idea. basically, if you try to do that, you just won't solve any long problems.

three-set venn problems are clearly longer than average, so it's no problem to take more than 2 minutes here... as long as you're not lost. (if you're lost, you should guess and move on, even if it's only been thirty seconds.)