Appendix C - Induction and the Binomial Theorem - Exercises - Page A15: 14

$(a+b)^6=a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6$ and $(a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4 $

We need to expand $(a+b)^6$ by using Pascal`s Triangle. Therefore, we have: $(a+b)^6=1a^6b^0+6a^5b^1+15a^4b^2+20a^3b^3+15a^2b^4+6a^1b^5+1a^0b^6 =a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6$ We need to expand $(a-b)^4$ by using Pascal`s Triangle. Therefore, we have: $[a+(-b)]^4=1a^4(-b^0)+4a^3(-b)^1+6a^2(-b)^2+4a^1(-b)^3+(-b)^4 =a^4-4a^3b+6a^2b^2-4ab^3+b^4 $