If set A={2,2,2,....,n times} , set B={3,3,3,3....m times} and set C

Question

If set A={2,2,2,....,n times} , set B={3,3,3,3....m times} and set C={11,11,11...k time} , then in terms of m,n and k , how many possible subsets from sets A,B, and C can be created ?

a) k(n+m+mn)+ k
b) (1+n+m+mn)(k+1)
c) k^2(mn+n/m)
d) kmn(k+m+n)
e) None of the above.

mikemcgarry · Answer

I'm happy to respond. :-)

This is simply a problem testing the Fundamental Counting Principle, which is discussed in this blog: https://magoosh.com/gmat/2012/gmat-quant-how-to-count/ In the subset, of course, order doesn't matter. Notice that, in technical terms, the wording is very sloppy: " the subsets of these three sets " is a very sloppy vague idea. Presumably, the writer means, " the subsets of the union of these three sets ." Assuming this is what is meant, then all that matters is: a) how many 2's are included? b) how many 3's are included? and c) how many 11's are included? For the number of 2, we could have zero 2's, or one 2, or two 2's, all the way up to n 2's. That's (n + 1) possibilities for the 2's. Similarly, (m + 1) possibilities for the 3's and (k + 1) possibilities for the 11's. We simply multiply these three numbers. NOTICE that one set, the set that includes no 2's, no 3's, and no 11's, is included. This is known in mathematics as the null set , sometimes called the empty set, a set with no members. Technically, this is a subset of every possible set, but that's a technical detail of set theory that goes well beyond what the GMAT would expect students to know. Even the the calculation is not that difficult, some of the technical aspects of this question are not in line with the GMAT's expectations. The number of subsets is (m + 1)(n + 1)(k + 1). The answer is not given in that form. Instead, the first two factors have been FOILed together: (mn + m + n + 1)(k + 1). I don't know the source. This looks like a question that was written by someone who knows a good deal of math but not necessarily a lot about the GMAT. Does all this make sense? Mike :-)

Lucky2783 · Answer

hi  mikemcgarry  ,

thanks for your detailed explanation . it makes sense. 
Is it not like a GMAT question? To be honest i created this question when i looked at prime factorization and #factors of product of Integers. 
if N=2^n*3^m*11^k then #factors (n+1)*(m+1)*(k+1) , here we have 1 as one of the factor which is {} null set in your explanation.

Thanks

mikemcgarry · Answer

Dear Lucky, My friend, don't underestimate what it take to write GMAT-worthy questions. Think about, say, high school math. A good Algebra One teacher does NOT have an understanding of mathematics that ends with Algebra One. A good Algebra One teacher should understand everything through Calculus and beyond, so that he is aware of all the deeper implications of all the Algebra One ideas. Much in the way way, with any material on the GMAT, you need to understand a much greater level of detail about the subject at hand. The actual questions on the GMAT test material that lies in a certain range of difficulty, but to write good GMAT questions, you need to understand all the material in the question at a level several years beyond the level of the difficulty level of the GMAT itself. It's true that ordinary GMAT test takers do not need to know anything at all about formal set theory. BUT, when you write a question that includes sets, you need to understand set theory at a much deeper level, so that there's no ambiguity. This issue of whether to include the null set as a subset of other sets---that opens up deep questions in set theory, and your question naively stumbles into this territory. The question you wrote is, in these respects, very different from the question: If T = (2^n)*(3^m)*(11^k), then how many factors does T have? Admittedly, that is a relatively easy question in that form. I understand that you were trying to make the question trickier by changing it to this form. Many of the poorly written math questions that appear on this site result from someone trying to make a relatively easy question trickier, sometimes in manner that the writer doesn't fully understand. Such a question falls tremendously short of the high standards of the GMAT. Questions on the GMAT are difficult, not simply because someone took an easy question and tried to make it tricky. The questions are deeply perceptive, and reflect an understanding of the typical patterns of thinking and typical oversights of test takers. The math questions on the GMAT reflect profound mathematical thinking, not just a bare understand of what's in the Math Review. It's very hard to write GMAT-worthy questions. Does all this make sense? Mike :-)

apolo · Answer

This question, in the way written, is indeed wrong: repetition of an element(s) does not change a set, i.e., {1,2,2}={1,2}={1,1,2}.

Hence, set A={2,2,2,...., n times}={2}; similarly set B={3} and set C={11}

Also the question is ambiguous. One interpretation is that it is asking the number of all possible subsets of the union of A, B and C. Another interpretation is that it asks the sum of possible subsets of sets individually.

mikemcgarry · Answer

Dear apolo , My friend, that's an excellent point, but even here, we have issues with the subtleties of set theory. Whether what you say stands or not depends very much on the conventions chosen for the sets in question: repeated elements could be allowed, and are allowed in some sets. That would have to be specified explicitly. For example, at the level of algebra, we could say that the "set" of solutions to 0 = (x^3) - 6(x^2) + 9x is {0, 3, 3}. In some mathematical contexts (e.g. where are the x-intercepts of the graph ?) we don't care about repeated elements, and the roots are simply {0, 3}. In other contexts (e.g. what are the algebraic factors of the expression ?) having all three roots is crucial, even if some are repeated. Context is everything. Of course, all this is well beyond the GMAT! At least implicitly, given that each set has no more than one distinct element, it would seem repeated elements have been allowed by the convention here. Nevertheless, I think the larger problem remains. Set theory is an enormously complex region, much more challenging than what folks need to know for the GMAT, and no one has any business making a GMAT problem about sets unless he understands sets extraordinarily well. Does all this make sense? Mike :-)

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If set A={2,2,2,....,n times} , set B={3,3,3,3....m times} and set C

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