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Let's say I have the following 10 distinct items: {a,b,c,d,e,f,g,h,i,j}.

How many combinations are there if I can choose UP TO 10 items and order does not matter and items cannot be repeated?

I'm thinking that it is simply the sum of all the C(n,r)s, correct?

r0 = 1 r1 = 10 r2 = 45 r3 = 120 r4 = 210 r5 = 252 r6 = 210 r7 = 120 r8 = 45 r9 = 10 r10 = 1

Thus, 1024 combinations???

I appreciate any help... These always trip me up!

TLile5453
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  • As there are only 10 elements on your set in the first place, what you're doing is counting the number of parts of the set. Essentially, how you do it is just go through each element and pick it, or don't. That leaves you with $2^{|X|}=2^{10}$. – Bcpicao Apr 22 '20 at 19:17
  • Great! Thanks, @Bcpicao. – TLile5453 Apr 22 '20 at 19:19

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That’s correct. It’s equivalent to asking how many subsets of a set with $m$ elements are possible with size no more than $n$. We can either choose $0$ elements, or $1$ element, or $2$, or $3$ and so on till $n$ elements. The answer would be then be $$\sum_{i=0}^n {m \choose i}$$

Vishu
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