If you're aware of the binomial theorem, you have that:
[tex](x-y)^6=\sum_{k=0}^{6}\binom{6}{6-k}x^k(-1)^{6-k}y^{6-k}[/tex]
[tex](x-y)^6=\sum_{k=0}^{6}\binom{6}{k}x^k(-1)^{k}y^{6-k}[/tex]
The fourth term is then the case [tex]k=3[/tex], from which:
[tex]c_{3}=\binom{6}{3}x^3(-1)^3y^3= \frac{6!}{3!(6-3)!} * (-x^3y^3)=- \frac{720}{36}x^3y^3=-20x^3y^3 [/tex]
