[tex](a+b)^9=(^9_0)a^9+(^9_1)a^8b^1+(^9_2)a^7b^2+(^9_3)a^6b^3+(^9_4)a^5b^4+(^9_5)a^4b^5+(^9_6)a^3b^6+(^9_7)a^2b^7+(^9_8)a^1b^8+(^9_9)b^9\\[/tex]
[tex](a+b)^9=\frac{9!}{0!*9!}a^9+\frac{9!}{1!*8!}a^8b+\frac{9!}{2!*7!}a^7b^2+\frac{9!}{3!*6!}a^6b^3+\frac{9!}{4!*5!}a^5b^4+\frac{9!}{5!*4!}a^4b^5+\frac{9!}{6!*3!}a^3b^6+\frac{9!}{7!*2!}a^2b^7+\frac{9!}{8!*1!}ab^8+\frac{9!}{9!*0!}b^9\\\\\underline{(a+b)^9=a^9+9a^8b+36a^7b^2+84a^6b^3+126a^5b^3+126a^4b^5+84a^3b^6+36a^2b^7+9ab^8+b^9}[/tex]
[tex]\text{Wiersz 9: }\\1, 9, 36, 84, 126, 126, 84, 36, 9, 1[/tex]
[tex](a+b)^{10}=(^{10}_0)a^{10}+(^{10}_1)a^9b^1+(^{10}_2)a^8b^2+(^{10}_3)a^7b^3+(^{10}_4)a^6b^4+(^{10}_5)a^5b^5+(^{10}_6)a^4b^6+(^{10}_7)a^3b^7+(^{10}_8)a^2b^8+(^{10}_9)a^1b^9+(^{10}_{10})b^{10}[/tex]
[tex](a+b)^{10}+\frac{10!}{0!*10!}a^{10}+\frac{10!}{1!*9!}a^9b^1+\frac{10!}{2!*8!}a^8b^2+\frac{10!}{3!*7!}a^7b^3+\frac{10!}{4!*6!}a^6b^4+\frac{10!}{5!*5!}a^5b^5+\frac{10!}{6!*4!}a^4b^6+\frac{10!}{7!*3!}a^3b^7+\frac{10!}{8!*2!}a^2b^8+\frac{10!}{9!*1!}ab^9+\frac{10!}{10!}b^{10}[/tex]
[tex]\underline{(a+b)^{10}=a^{10}*10a^9b+45a^8b^2+120a^7b^3+210a^6b^4+252a^5b^5+210a^4b^6+120a^3b^7+}\\\underline{45a^2b^8+10ab^9+b^{10}}[/tex]
[tex]\text{Wiersz 10: }\\1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1[/tex]
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[tex]\text{Wspolczynnik dwumianowy: }(^n_k)=\frac{n!}{k!(n-k)!}[/tex]
[tex]\text{Wzor na Dwumian Newtona: }(a+b)^n : \\\\(a+b)^n=a^nb^0+(^n_{1})a^{n-1}b^1+(^n_{2})a^{n-2}b^{2}+(^n_3)a^{n-3}b^3+...+(^n_{n-1})a^1b^{n-1}+(^n_n)a^0b^n[/tex]
