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20 Jan 2023, 01:48
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woohoo921
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
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
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?
