� Properties of the combination coefficient

�n� n!

1 . � � _� �r � r!( n � r)!

�n� � �: �r

�

� The binomial theorem : �n� � n� � n� � n � � n�

( a � b)

� � �a b � � �a b � � �a b � � � �a b � � �a b �0� �1� � 2� � n�1� � n�

�

n n 0 n�1 1 n�2 2 1 n�1 0 n

�n� _a � � n n�r r

� Extended binomial theorem for x � 1:

�n� -th term � _{� �a} �r

� n � n�

2 � n�

3

( 1 � x) �1� nx � � � x � � � x � � 2�

� 3�

n ( n)( n �1 ) 2 ( n)( n �1 )( n � 2 ) 3

( 1 � x) �1� nx � x � x � ( 2 )( 1 ) ( 3 )( 2 )( 1 )

n�r b r

Mathematical Induction

� Steps of proving by mathematical induction : 1 . Prove that the statement Pn ( ) is true when n � 1

2 . Assume that Pn ( ) is true when n

� k 3 . Prove that the statement Pn ( ) is true when n�k�

1 4 . Conclude that Pn ( ) is true for all positive integer n

� Types of mathematical induction : 1 . General case 2 . Divisibility www . seprodstore . com

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