n C n – a . b n – + n C n b n Proof The proof is obtained by applying principle of mathematical induction. Let the given statement be P( n ) : ( a + b ) n = n C a n + n C a n – b + n C a n – b + ...+ n C n – a . b n – + n C n b n For n = , we have P ( ) : ( a + b ) = C a + C b = a + b Thus, P ( ) is true.
Suppose P ( k ) is true for some positive integer k , i.e. ( a + b ) k = k C a k + k C a k – b + k C a k – b + ...+ k C k b k ... ( ) We shall prove that P( k + ) is also true, i.e., ( a + b ) k + = k + C a k + + k + C a k b + k + C a k – b + ...+ k + C k + b k + Now, ( a + b ) k + = ( a + b ) ( a + b ) k = ( a + b ) ( k C a k + k C a k – b + k C a k – b +...+ k C k – ab k – + k C k b k ) [from ( )] = k C a k + + k C a k b + k C a k – b +...+ k C k – a b k – + k C k