This is an age old question. Which actually does (I think even cannot have) definite answer. Because, first you need to define what you mean by a cluster and so on. Famous saying in this regard is that "cluster is in the eye of a beholder", it is easy to construct examples where somebody could see one cluster but somebody else more than one.

This being said, MDL (minimum description length) principle would lead you to device a (IMHO) a most principled way clustering cost function. Which by optimizing you could the find the cluster assignments and number of clusters simultaneously. For multinomial data you can see following: P.Kontkanen, P.Myllymäki, W.Buntine, J.Rissanen, H.Tirri, An MDL Framework for Data Clustering. In Advances in Minimum Description Length: Theory and Applications, edited by P. Grünwald, I.J. Myung and M. Pitt. The MIT Press, 2005.

The intuitively appealing idea behind MDL clustering is that by clustering you create a model of the data. So assumption is that very good model is such which lets you compress the data well.

Anyway MDL might not be easy to apply, if you are looking for a practical way to detect the number of clusters. BIC (Bayesian information criteria) and F-ratio have proven to work ok in practice.

BIC formulation can be found in: http://staff.utia.cas.cz/nagy/skola/Projekty/Classification/Xmeans.pdf

This is an age-old question, which actually does not have (I think even cannot have) a definite answer, because first you need to define what you mean by a cluster and so on. A famous saying in this regard is that "cluster is in the eye of a beholder". It is easy to construct examples where somebody could see one cluster, but somebody else more than one.

This being said, the MDL (minimum description length) principle would lead you to devise (IMHO) a clustering cost function in a most principled way, which by optimizing you could the find the cluster assignments and number of clusters simultaneously. For multinomial data you can see following: P.Kontkanen, P.Myllymäki, W.Buntine, J.Rissanen, H.Tirri, An MDL Framework for Data Clustering. In Advances in Minimum Description Length: Theory and Applications, edited by P. Grünwald, I.J. Myung and M. Pitt. The MIT Press, 2005.

The intuitively-appealing idea behind MDL clustering is that by clustering you create a model of the data. So the assumption is that a very good model is one that lets you compress the data well.

Anyway MDL might not be easy to apply, if you are looking for a practical way to detect the number of clusters. BIC (Bayesian information criteria) and the F-ratio have proven to work OK in practice.