Permutated Pooled Pascal (Power of 7)

Algebra Level 3

\[ \begin{eqnarray} && { (1+2+3+4+5+6+7) }^{ 1 } \\ & +& 7 { (1+2+3+4+5+6) }^{ 2 } \\ & +&21{ (1+2+3+4+5) }^{ 3 } \\ & +&35{ (1+2+3+4) }^{ 4 } \\ & +&35{ (1+2+3) }^{ 5 } \\ & +&21{ (1+2) }^{ 6 } \\ & +&7 { (1) }^{ 7 } \\ & =& \ ? \end{eqnarray} \]

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