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Solutions from a student (dragged) 23

Solutions from a student (dragged) 23 - Part(c Consider the...

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Part (c): Consider the following manipulations of a binomial like sum n j = i n j j i x j - i y n - i - ( j - i ) = n j = i n i n - i j - i x j - i y n - j = n i n j = i n - i j - i x j - i y n - j = n i n - i j =0 n - i j x j y n - ( j + i ) = n i n - i j =0 n - i j x j y n - i - j = n i ( x + y ) n - i . In summary we have shown that n j = i n j j i x j - i y n - j = n i ( x + y ) n - i for i n Now let x = 1 and y = - 1 so that x + y = 0 and using these values in the above we have n j = i n j j i ( - 1) n - j = 0 for i n . Problem 15 (the number of ordered vectors) As stated in the problem we will let H k ( n ) be the number of vectors with components x 1 , x 2 , · · · , x k for which each x i is a positive integer such that 1 x i n and the x i are ordered i.e. x 1 x 2 x 3 · · · x n Part (a): Now H 1 ( n ) is the number of vectors with one component (with the restriction on its value of 1 x 1 n ). Thus there are n choices for x 1 so H 1 ( n ) = n . We can compute
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