(1
+
x)
7
=
(
1
+
x
)
(
1
+
x
)
(
1
+
x
)
(
1
+
x
)
(
1
+
x
)
(
1
+
x
)
(
1
+
x
)
Conclusion
$$ (1+x)^7 = \sum_{A\subseteq\{1,\dotsc,7\}}x^{|A|} = \sum_{k=0}^7\left(\begin{matrix}7 \\ k\end{matrix}\right)x^k = 1 + 7x + 21 x^2 + 35 x^3 + 35x^4 + 21 x^5 + 7 x^6 + x^7 $$
