"No more than 3" means he doesn't have to bring _exactly_ 3 of his 7 lawyers: he can also bring only 0, 1 or 2 of them:. This gives you \begin{align} \text{number} &= \binom{7}{0} + \binom{7}{1} + \binom{7}{2} + \binom{7}{3}\\\ &= 1 + 7 + 21 + 35\\\ & = 64 \end{align}