The joint distribution function of the number of renewals in an ordinary renewal process is derived using a result from multiple integrals. The well-known Poisson process is generated. It is also shown that the distribution of the number of renewals in an ordinary renewal counting process is a solution to a set of birth equations with non-homogeneous state-dependent rates which are explicitly derived. This provides an example of a process which, although in general, does not satisfy the Chapman Kolmogorov equations, has nevertheless an unconditional distribution which satisfies a set of birth equations