ADVANCED EXERCISES - Expanding Binomials

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(1) Use the Binomial Theorem to show that

 n + n + n + ... +  n  + n  =  2n .
 0   1   2         n-1   n

( Click On HINT To Get Some Help , Click On SOLUTION To See The Answer )


(2) Use the Binomial Theorem to show that

 n - n + n - ... + (-1)k n + ... + (-1)n n  =  0 .
 0   1   2               k              n

( HINT , SOLUTION )


(3) Use the Binomial Coefficient Formula and 0! = 1 to prove the following identities:

         (i)  n  =  1 .
              0


        (ii)  n  =  n .
              1


       (iii)  n  =   n   for 0  r  n .
              r     n-r

( HINT , SOLUTION )

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