Show that $$(1-t^6)^6 = \sum_{j=0}^{6} {6 \choose j} (-1)^jt^{6j} $$
Using the binomial theorem $(x+y)^n = \sum_{j=0}^{n} {n \choose k} x^{n-k} y^{k}$, I get $$ \text{Let } \\x = 1 \\y = (-t^6) \\ n=6 $$ then $(1-t^6)^6 = \sum_{j=0}^{6} {6 \choose j} (1)^{6-j}\cdot{-t^{6j}} \text{ but not } \sum_{j=0}^{6} {6 \choose j} (-1)^jt^{6j}$ .