n    Binomial    Expanded Binomial
 
  1 (a + b)1 a + b
 
  2 (a + b)2 a2 + 2ab + b2
 
  3 (a + b)3 a3 + 3a2b + 3ab2 + b3
 
  4 (a + b)4 a4 + 4a3b + 6a2b2 + 4ab3 + b4
 
  5 (a + b)5 a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 
 
 
The expression of the binomial theorem is (a + b)n = 1 where a and b are the respective probabilities of the two alternative outcomes and n equals the number of trials.

For example, what is the probability (p) of having two boys and two girls in a family? Let a and b be the probability of having a boy (1/2) and a girl (1/2), respectively. For n = 4, use 6a2b2 in the expanded term. 

p = 6a2b2
  = 6(1/2)2(1/2)2
  = 6(1/2)4
  = 6(1/16)
  = 6/16
  = 3/8