  This paper presents a multivariate normal integral expression for the joint
survival function of the cumulated components of any multinomial random vector.
This result can be viewed as a multivariate analog of Equation (7) from Carter
& Pollard (2004), who improved Tusn\'ady's inequality. Our findings are based
on a crucial relationship between the joint survival function of the cumulated
components of any multinomial random vector and the joint cumulative
distribution function of a corresponding Dirichlet distribution. We offer two
distinct proofs: the first expands the logarithm of the Dirichlet density,
while the second employs Laplace's method applied to the Dirichlet integral.
