Question 614924
how many subsets are possible in a set with 3 elements.
let's see.
suppose the elements are a,b,c
you can have:
abc
ab
ac
bc
a
b
c
how many subsets are possible in a set with 4 elements.
let's see.
suppose the elements are a,b,c,d
you can have:
abcd
abc
abd
acd
bcd
ab
ac
ad
bc
bd
cd
a
b
c
d
we don't count the null set (no elements in it).
the number of subsets when you have 3 elements becomes:
3c1 + 3c2 + cc3
the number of subsets when you have 4 elements becomes:
4c1 + 4c2 + 4c3 + 4c4
the number of subsets when you have 55 elements becomes:
55c1 + 55c2 + 55c3 + ......... + 55c54 + 55c55
that's a big number.
i used excel to calculate it.
here's the results:
<pre>
	total >>>>>>>>>>>>	3.60288E+16
n	x	ncx
55	1	55
55	2	1485
55	3	26235
55	4	341055
55	5	3478761
55	6	28989675
55	7	202927725
55	8	1217566350
55	9	6358402050
55	10	29248649430
55	11	1.19654E+11
55	12	4.3873E+11
55	13	1.45118E+12
55	14	4.35355E+12
55	15	1.18997E+13
55	16	2.97493E+13
55	17	6.82483E+13
55	18	1.4408E+14
55	19	2.80576E+14
55	20	5.05037E+14
55	21	8.41729E+14
55	22	1.30085E+15
55	23	1.86644E+15
55	24	2.48859E+15
55	25	3.08585E+15
55	26	3.5606E+15
55	27	3.82435E+15
55	28	3.82435E+15
55	29	3.5606E+15
55	30	3.08585E+15
55	31	2.48859E+15
55	32	1.86644E+15
55	33	1.30085E+15
55	34	8.41729E+14
55	35	5.05037E+14
55	36	2.80576E+14
55	37	1.4408E+14
55	38	6.82483E+13
55	39	2.97493E+13
55	40	1.18997E+13
55	41	4.35355E+12
55	42	1.45118E+12
55	43	4.3873E+11
55	44	1.19654E+11
55	45	29248649430
55	46	6358402050
55	47	1217566350
55	48	202927725
55	49	28989675
55	50	3478761
55	51	341055
55	52	26235
55	53	1485
55	54	55
55	55	1
</pre>
the total number of possible subsets looks like it's 3.60288 * 10^16.
Each one of the subsets is unique in that it contains at least 1 element in it that's unique from any other subset with the same number of elements in it.
i won't swear that this is correct, but it's based on a reasonable assumption.
look at the set with 3 elements in it and you'll see that each of the subsets is unique.
the subset with 3 elements in it is abc which is the set itself.  
every set is a subset of itself.
the subsets with 2 elements in them are unique.
they contain:
ab
ac
bc
the subsets with 1 element in them are unique.
they contain:
a
b
c
we cover all of the subsets by including all subsets that have 1 less element in them then the previous set of subsets.
this is my guess.
like i say, i won't swear to it, but i think it's a reasonable assumption.
without any precedence to guide me, i'd say this is my best guess.
there are intersections with the other subsets.
for example:
the set ab and the set ac have a common element of a.
the set ac and the set bc have a common element of c.
this is to be expected.
they are still unique subsets because they each have an element in them that is not in the other set.