is given bynl=knl=nn+nn-1+nn-2+. . .+nk+ 1+nk.Problem 6 (counting the number of increasing vectors)If the first componentx1were to equaln, then there is no possible vector that satisfies theinequalityx1< x2< x3< . . . < xkconstraint.If the first componentx1equalsn-1then again there are no vectors that satisfy the constraint.The first largest value thatthe componentx1can take on and still result in a complete vector satisfying the inequalityconstraints is whenx1=n-k+1 For that value ofx1, the other components are determinedand are given byx2=n-k+ 2,x3=n-k+ 3, up to the value forxkwherexk=n.This assignment providesonevector that satisfies the constraints. Ifx1=n-k, then wecan construct an inequality satisfying vectorxby assigning thek-1 other componentsx2,x3, up toxkby assigning the integersn-k+ 1, n-k+ 2, . . . n-1, nto thek-1components. This can be done ink1ways. Continuing if
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