Seven persons  $P_1, P_2,\dots \dots \dots , P_7$ initially seated at chairs $C_1, C_2\dots \dots \dots , C_7$, respectively. They all left there chairs simultaneously for hand wash. Now in how many ways they can again take seats such that no one sits on his own seat and $P_1$ sits on $C_2$ and $P_2$ sits on $C_3?$

A rectangle with sides $2m-1$ and $2n-1$ is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is

Find the number of all rational numbers $\frac{m}{n}$ such that:

(i) $0<\frac{m}{n}<1$

(ii) $m$ and $n$ are relatively prime

(iii) $m·n=25!$