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How to prove $3^n = \sum_{0 \leq j \leq i \leq n} $ $ n \choose i$ $ i \choose j$ using $3^n = \sum_{0 \leq i \leq n} 2^i$ $n \choose i$

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3  
$2 = 1+1$. :-) Also, shouldn't it be $j \leq i$? –  WimC Nov 19 '12 at 16:51
    
@WimC yes typo, thanks –  xiamx Nov 19 '12 at 16:52

2 Answers 2

up vote 11 down vote accepted

$$ \sum_{0 \leq j \leq i \leq n} {n \choose i} {i \choose j}=\sum_{0 \le i \le n} {n \choose i} \sum_{0 \le j \le i}{i \choose j}=\sum_{0 \le i \le n} {n \choose i} 2^i=3^n. $$

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Count cardinality of $S = \{(A,B):B \subseteq A \subseteq \left\{1,2,\dots,n\right\}\}$ in two different ways:

Way 1. Each element of $\{1,2,\dots,n\}$ can either be in $A$ and $B$, only in $A$, or in none of $A$ and $B$, so $|S|=3^n$.

Way 2. If $|A|=i$ and $|B|=j$, then there are $n \choose i$ options for $A$ and $i \choose j$ options for $B$, therefore $|S|=\sum_{j \leq i} {n \choose i} {i \choose j}$.

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