Combinatorics with Pascal's Triangle

$a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5$

$a^5 + 4a^4b + 3a^3b^2 + 3a^2b^3 + 4ab^4 + b^5$

$a^5 + b^5$

$a^5 - 5a^4b + 25a^3b^2 + 25a^2b^3 - 5ab^4 + b^5$