What I have been doing.

For the past few minutes, anyway...

(a+b)^7
= (a+b)(a+b)(a+b)(a+b)(a+b)(a+b)(a+b)
= (a^2 + ab + ab + b^2)(a+b)(a+b)(a+b)(a+b)(a+b)
= a(a^2 + ab + ab + b^2) + b(a^2 + ab + ab + b^2)(a+b)(a+b)(a+b)(a+b)
= a^3 + a^2b + a^2b + ab^2 + a^2b + ab^2 + ab^2 + b^3 (a+b)(a+b)(a+b)(a+b)
= (a^3 + 3a^2b + 3ab^2 + b^3)(a+b)(a+b)(a+b)(a+b)
= a(a^3 + 3a^2b + 3ab^2 + b^3) + b(a^3 + 3a^2b + 3ab^2 + b^3)(a+b)(a+b)(a+b)
= (a^4 + 3a^3b + 3a^2b^2 + ab^3 + a^3b + 3a^2b^2 + 3ab^3 + b^4)(a+b)(a+b)(a+b)
= (a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4)(a+b)(a+b)(a+b)
= a(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4) + b(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4)(a+b)(a+b)
= (a^5 + 4a^4b + 6a^3b^2 + 4a^2b^3 + ab^4 + a^4b + 4a^3^b2 + 6a^2b^3 + 4ab^4 + b^5)(a+b)(a+b)
= (a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5)(a+b)(a+b)
= a(a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5) + b(a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5)(a+b)
= (a^6 + 5a^5b + 10a^4b^2 +10a^3b^3 + 5a^2b^4 + ab^5 + a^5b + 5a^4b^2 + 10a^3b^3 + 10a^2b^4 + 5ab^5 + b^6)(a+b)
= (a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6)(a+b)
= a(a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6) + b(a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6)
= a^7 + 6a^6b + 15a^5b^2 + 20a^4b^3 + 15a^3b^4 + 6a^2b^5 + a^6b + a^6b^2 + 6a^5b^2 + 15a^4b^3 + 20a^3b^4 + 15a^2b^5 + 6ab^6 + b^7
= a^7 + 7a^6b + 21a^5b^2 + 35a^4b^3 + 35a^3b^4 + 21a^2b^5 + 7ab^6 + b^7