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20 Jan 2023, 01:48
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woohoo921
JDPB7
The subsets of the set {w, x, y} are {w}, {x}, {y}, {w, x}, {w, y}, {x, y}, {w, x,y}, and { } (the empty subset). How many subsets of the set {w, x, y, z} contain w ?
(A) Four
(B) Five
(C) Seven
(D) Eight
(E) Sixteen
(A) Four
(B) Five
(C) Seven
(D) Eight
(E) Sixteen
KarishmaB
If you have time, I would be so appreciative for your insights on this problem. Is the formula that Bunuel is using standardized and can be used for future problems? How do you come up with this formula? Thank you and Happy Holidays
(Total number of sets) - (number of sets without w) = (number of sets with w)
2^4 - 2^3 = 8.
Hello woohoo921,
Yes, the formula used by Bunuel is a general formula.
Formula:
If a set S has ‘n’ elements, then the total number of subsets of S is 2n
Derivation:
- Let’s consider a random set S with n elements: {\(e_1\), \(e_2\), \(e_3\), ….. \(e_n\)}
- First observe that subsets are formed by using some elements and not using others from the main set.
- For example: {1, 2} is a subset of {1, 2, 3} for which we kept both 1 and 2 in but 3 out. Similarly, {3} is a subset of {1, 2, 3} for which we kept only 3 in and kept both 1 and 2 out.
- So, the main decision in creating any subset is on which elements to keep and which not to keep in the subset. In other words, for each element of the main set, we just need to make one out of two choices – keep this element (IN) OR do not keep this element (OUT).
- Coming back to set S, each element from all n has 2 choices: IN or OUT. We need to see the total number of ways in which we could take this decision for all n elements. That will give us the total number of subsets of S. (It now looks like a P&C question!)
- Thus, total number of subsets would depend on the different ways n places can be filled with either IN or OUT.
- The first place can be filled in 2 ways, second place can be filled in 2 ways, third place can be filled in 2 ways and so on and so forth.
- Total number of subsets = 2*2*2*2*…. n times
- = \(2^n\)
And that’s it!
Example:
Let’s take a very small set: {1, 2}.
We can quickly list and know that there are a total of 4 subsets – {1}, {2}, {1, 2}, {}.
Now, we’ll just verify that the formula gives us the same answer.
Since n = 2, total number of subsets = \(2^n\) = \(2^2\) = 4. That’s how simple the formula makes it!
Hope this helps!
Best Regards,
Ashish
Quant Expert, e-GMAT
31 Mar 2023, 06:45
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JDPB7
The subsets of the set {w, x, y} are {w}, {x}, {y}, {w, x}, {w, y}, {x, y}, {w, x,y}, and { } (the empty subset). How many subsets of the set {w, x, y, z} contain w ?
(A) Four
(B) Five
(C) Seven
(D) Eight
(E) Sixteen
(A) Four
(B) Five
(C) Seven
(D) Eight
(E) Sixteen
1(selecting a single letter set)+3C1(Selecting two letters set->W is fixed & selecting the second letter from remaining 3 letters)+3C2(Selecting three letters set->W is fixed & selecting the other two letters from remaining 3 letters)+1(set with all four letters)
=8
Is it a right way to solve?
30 Jan 2024, 03:14
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Bunuel
thinkpad18
EMPOWERgmatRichC
Hi JDPB7,
The prompt itself literally tells you how to go about answering this question. You're asked for the total number of available subsets that contain W, and you're shown the 'definition' of what makes up a subset. With that knowledge, you should be able to list them all out (and there can't be that many, since the answers don't go any higher than 16).
If you try to create the list, then how many options do you come up with?
GMAT assassins aren't born, they're made,
Rich
The prompt itself literally tells you how to go about answering this question. You're asked for the total number of available subsets that contain W, and you're shown the 'definition' of what makes up a subset. With that knowledge, you should be able to list them all out (and there can't be that many, since the answers don't go any higher than 16).
If you try to create the list, then how many options do you come up with?
GMAT assassins aren't born, they're made,
Rich
Is there a way to get to 8 from a combination method? I got 16 from finding the total combinations but how do I parse out which sets contain W and which ones do not? Just divide by 2 because the set has it or doesn't have it?
Obviously straight-listing works too!
(Total number of sets) - (number of sets without w) = (number of sets with w)
2^4 - 2^3 = 8.
Thanks. Please can you help me understand how would we solve something like- a set contains 6 elements, how many ways exist to choose a subset of 3 from this set?
