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Prove that underset(rles)(underset(r=0)o...

Prove that `underset(rles)(underset(r=0)overset(s)(sum)underset(s=1)overset(n)(sum))""^(n)C_(s) ""^(s)C_(r)=3^(n)-1`.

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`underset(rles)(underset(r=0)overset(s)(sum)underset(s=1)overset(n)(sum)).^(n)C_(s).^(s)C_(r)=underset(s=1)overset(n)sum.^(n)C_(s)(.^(s)C_(0)+.^(s)C_(1)+.^(s)C_(2)+"....."+.^(s)C_(s))`
`= underset(s=1)overset(n)sum.^(n)C_(s)2^(s)`
`= underset(s=0)overset(n)sum.^(n)C_(s)2^(s)-.^(n)C_(0)2^(0)`
`= (1+2)^(n)-1`
`= 3^(n) - 1`
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