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Solutions from a student (dragged) 17

# Solutions from a student (dragged) 17 - is given by n l =k...

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is given by n l = k n l = n n + n n - 1 + n n - 2 + . . . + n k + 1 + n k . Problem 6 (counting the number of increasing vectors) If the first component x 1 were to equal n , then there is no possible vector that satisfies the inequality x 1 < x 2 < x 3 < . . . < x k constraint. If the first component x 1 equals n - 1 then again there are no vectors that satisfy the constraint. The first largest value that the component x 1 can take on and still result in a complete vector satisfying the inequality constraints is when x 1 = n - k +1 For that value of x 1 , the other components are determined and are given by x 2 = n - k + 2, x 3 = n - k + 3, up to the value for x k where x k = n . This assignment provides one vector that satisfies the constraints. If x 1 = n - k , then we can construct an inequality satisfying vector x by assigning the k - 1 other components x 2 , x 3 , up to x k by assigning the integers n - k + 1 , n - k + 2 , . . . n - 1 , n to the k - 1 components. This can be done in k 1 ways. Continuing if
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