Binomial expansion
[tex](a+b)^n=\sum ^n_{k=0}nC_k\cdot a^{n-k}\cdot b^k[/tex]The coefficient of each term is:
[tex]_nC_k[/tex]That is, the number of combinations of k objects from a set with n objects
In the binomial:
[tex](a+b)^{12}[/tex]the value of n is n = 12. In the fourth term, k = 3 (remember that k starts at zero). Therefore, the coefficient of the 4th term is:
[tex]_{12}C_3=\frac{12!}{(12-3)!\cdot3!}=\frac{12!}{9!\cdot3!}=\frac{12\cdot11\cdot10\cdot9!}{9!\cdot3\cdot2\cdot1}=\frac{1320}{6}=220[/tex]