Seven scientists$S_1 , S_2 ,....S_7$ are invited to deliver one lecture each in a conference. The number of ways all the seven lectures can be arranged such that the lecture of $S_1$ is prior to that of $S_3$ and the lecture of$S_3$ is prior to that of $S_7$ is

$$\begin{gathered} 7 \mathrm{scientists} {\mathrm{S}}_1,{\mathrm{S}}_2\dots \dots {\mathrm{S}}_7 \mathrm{are} \mathrm{invited} \mathrm{to} \mathrm{deliver} \mathrm{lectures} \mathrm{consider} 7 \mathrm{places}. \\ \mathrm{Arrange} {\mathrm{S}}_2, {\mathrm{S}}_4, {\mathrm{S}}_5, {\mathrm{S}}_6 \mathrm{in} 7 \mathrm{places} \mathrm{in} 7{\mathrm{p}}_4 \mathrm{ways}. \\ \mathrm{In} \mathrm{the} \mathrm{remaining} 3 \mathrm{places} {\mathrm{S}}_1,{\mathrm{S}}_3,{\mathrm{S}}_{7 }\mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} \mathrm{are} \mathrm{way}. \\ \mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{ways} =7{\mathrm{p}}_4\left(1\right) \\ =\frac{7!}{3!}=7\times 5\times 4\times 3\times 2\times 1 \\ =840 \end{gathered}$$