In a d-dimensional convex body K, for nd+1, random points X0,,Xn1 are chosen according to the uniform distribution in K. Their convex hull is a random (n1)-simplex with probability 1. We denote its (n1)-dimensional volume by VK[n]. The k-th moment of the (n1)-dimensional volume of a random (n1)-simplex is monotone under set inclusion, if KL implies that the k-th moment of VK[n] is not larger than that of VL[n]. Extending work of Rademacher (Mathematika 58:7791, 2012) and Reichenwallner and Reitzner (Mathematika 62:949958, 2016), it is shown that for nd, the moments of VK[n] are not monotone under set inclusion. As a consequence, the nonmonotonicity of the expected surface area of the convex hull of nd+1 uniform random points in a d-dimensional convex body follows.