How many subsets of {2, 3, 4, 5, 6, 7, 8, 9} contain at least one prim

Total no. of subsets from {2, 3, 4, 5, 6, 7, 8, 9} = 2^8 = 256 --->> A
Total no. of subsets that do not contain any prime from {4, 6, 8, 9} = 2^4 = 16 --->> B
Total no. of subsets that contain at least one prime = A - B = 240

For each number, we can decide whether the Number is “IN” a given set or “OUT” of the given set.


Total Number of Sets Possible:


2 ——- in or out——— 2 options
And
3 ——- in or out ———- 2 options
And
4——- in or out ———- 2 options

.....
9 —— in or out ——— 2 options
_____________________

This gives us (2)^8 possibilities (which includes all the subsets that have one or two or three or four ....up to all eight of the numbers)

However, this also includes the 1 “empty subset” in which no numbers are included in the subset. So we need to subtract this one possibility out.

Total possible sets = (2)^8 - 1 = 255


From the total possible subsets, we need to Remove all of the cases that do NOT include at least 1 Prime number. These will be:

2——out

3——out

4 ——-IN or OUT —— 2 options

5——-out

6—— IN or OUT ——- 2 options

7——out

8—— IN or OUT ——2 options

9—— IN or OUT ——- 2 options
________________

(2)^4 subsets

However, again we need to remove the 1 possibility of the “empty subset” in which no numbers are included (they are all “out”)

(2)^4 - (1) = 15


255 - 15 = 240 subsets that include at least 1 prime number

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should not the answer be 239 ? As in one case we are considering all the 4 prime and all the 4 non-prime numbers, which is not a subset of the set mentioned.

 
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No, becaseu a set is a subset of itself. Hence, {2, 3, 4, 5, 6, 7, 8, 9} is a subset of {2, 3, 4, 5, 6, 7, 8, 9}.

Hi Bunuel,

Although I practiced 95% of the topics in Quant I found that I have a little gap in this topic.
Do you have an explanatory post on this topic? of the concept "Subsets" - computing / formulas / general idea behind it?

Thanks!