exercisespascal.pdf - Exercises Pascals Triangle Some of...

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Exercises: Pascal’s Triangle Some of these exercises use the “Hockey Stick Identities” n i = k ( i k ) = ( n + 1 k + 1 ) , n i = k ( i i - k ) = ( n + 1 n - k ) 1. Show by induction that the binomial coefficients are “bimodal”, that is, that as a function of k , the value of ( n k ) increases to the middle, and then decreases. 2. Show directly, not by induction, that if that for k n that ( 2 n k - 1 ) < ( 2 n k ) by showing that ( 2 n k - 1 ) / ( 2 n k ) < 1. 3. Show that for any finite set that the number of subsets of even cardinality is equal to the number of subsets of odd cardinality. 4. Show that for all n n i =0 ( n i )( 2 n n + i ) = ( 3 n n ) 5. Show that for all n 2 n i =0 ( 2 n i )( 3 n n + i ) = ( 5 n 2 n ) 6. Find constants a , b , and c so that i 2 = a ( i 0 ) + b ( i 1 ) + c ( i 2 ) . Use this expression and the Hockey Stick Identities to derive our formula for n i =1 i 2 7. Find constants
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