Let \(I\) be an ideal of height \(h>0\) in a local ring \((R, {\mathfrak m})\) and assume that \(R\) and \(R/I\) are Cohen-Macaulay. Recall that the mixed multiplicities \(e_j({\mathfrak m}\mid I)\) are defined by \[ \text{length} ({\mathfrak m}^r I^s/{\mathfrak m}^{r+1} I^s)=\sum_{i+j=d-1} e_j({\mathfrak m}\mid I)r^is^j +\text{ lower order terms}, \] where \(r,s\) are sufficiently large. The main result of this note is that the minimal number of generators \(\mu(I)\) satisfies bounds \(\mu(I)\leq h-q+(q-1) e(R/I)+e_{h-q} ({\mathfrak m}\mid I)\) for \(q=0,\dots,h\), where \(e(R/I)\) is the usual multiplicity of \(R/I\). The authors show that this inequality easily implies and even generalizes several other bounds for the minimal number of generators previously known in the literature.