The probability of the event of the multinomial distribution is given by the formula
Two methods for this exist. One is based on Moran's inequality, which is not presented in its sharper version in textbooks like Feller  or Johnson- Kotz-Balakrishnan . It is slow for large values of the number of the . By far more effective is the idea of Finucan  to consider the step-function
By a similar method the modes of a hypergeometric distribution can be calculated.
|||W. Feller , An Introduction to Probability Theory and its Applications. J. Wiley 1950, 1957 and 1968.|
|||H.M. Finucan , The mode of a Multinomial Distribution. Biometrica 51, 513-517.|
|||N.L. Johnson , S. Kotz and N. Balkrishnan, Discrete Multivariate Distributions. J. Wiley 1997.|