Number of ways of choosing at least $k$ objects out of $n$

Question

Suppose you have three distinct items $a$, $b$, $c$. You want to find how many unique sets you can get by choosing at least one item. For example, $\{a\}$ would form a unique set, and $\{a, b\}$ would form a unique set. Of course you can't choose more than three items but you have to choose at least one. What is the formula for this in the general case?

At first I tried the combination formula but I realized that doesn't apply because $n$ choose $k$ requires you choose exactly $k$ items. So what is the formula for fixed $n$ but varied $k$ (given max for $k$)?

score 2 · Accepted Answer · answered Jul 08 '15 at 02:07

2

Well, to get the number of ways of choosing at least $k$ objects from a set of $n$, the formula $$\sum_{i=k}^n\binom{n}{i}=\underbrace{\binom{n}{k}}_{\substack{\text{ways of choosing}\\\text{exactly }k}}+\underbrace{\binom{n}{k+1}}_{\substack{\text{ways of choosing}\\\text{exactly }k+1}}+\;\;\cdots\;\;+\underbrace{\binom{n}{n}}_{\substack{\text{ways of choosing}\\\text{exactly }n}}$$ works, but maybe it's not "closed-form" enough for your preferences?

answered Jul 08 '15 at 02:07

Zev Chonoles

129,973

1

Oh ok that works too. I thought there would be a way to compute it without adding it all up. – Josh Peck Jul 08 '15 at 02:11
Unfortunately, there's no good "closed-form" expression for this - take a look at this math.SE thread. – Zev Chonoles Jul 08 '15 at 02:12
Thanks! Also, I'm new to this terminology, but would the expression you gave me be considered closed-form because we know the number of operations need to compute the exact value? – Josh Peck Jul 08 '15 at 02:17
I suppose, but it's sort of a hazily-defined word to begin with :) Take a look at this math.SE thread for example. – Zev Chonoles Jul 08 '15 at 02:20

Number of ways of choosing at least $k$ objects out of $n$

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