The average monthly income of $A$ and $B$ is $\text{Rs. }1400$, that of $B$ and $C$ is $\text{Rs. }1560$ and of $A$ and $C$ is $\text{Rs. }1440$. The monthly income of $C$ is:

Let the monthly incomes of $A$, $B$, and $C$ be represented by $A$, $B$, and $C$ respectively.
From the given information:

1. $\frac{A+B}{2}=1400⟹A+B=2800$
2. $\frac{B+C}{2}=1560⟹B+C=3120$
3. $\frac{A+C}{2}=1440⟹A+C=2880$

Adding all three equations:

$$2(A+B+C)=2800+3120+2880=8800$$

$$A+B+C=4400$$

To find the monthly income of $C$, subtract $(A+B)$ from $(A+B+C)$:

$$C=(A+B+C)-(A+B)$$

$$C=4400-2800=1600$$

Thus, the monthly income of $C$ is $\text{Rs. }1600$.