To find the profit share of each partner, we need to calculate the ratio of the product of their invested capital and the duration of their investment.

Let the total capital be $K$ and the total duration be $T$.

1. Investment and Duration of $A$:

• Capital invested by $A$ = $\frac{1}{3}K$
• Duration of $A$'s investment = $\frac{1}{3}T$
• Equivalent investment of $A$ = $\frac{1}{3}K \times \frac{1}{3}T = \frac{1}{9}KT$

2. Investment and Duration of $B$:

• Capital invested by $B$ = $\frac{1}{4}K$
• Duration of $B$'s investment = $\frac{1}{4}T$
• Equivalent investment of $B$ = $\frac{1}{4}K \times \frac{1}{4}T = \frac{1}{16}KT$

3. Investment and Duration of $C$:

• Capital invested by $C$ = Remaining capital = $K - \left(\frac{1}{3}K + \frac{1}{4}K\right) = K - \frac{7}{12}K = \frac{5}{12}K$
• Duration of $C$'s investment = Whole duration = $T$
• Equivalent investment of $C$ = $\frac{5}{12}K \times T = \frac{5}{12}KT$

4. Ratio of Profits: The profit-sharing ratio of $A$, $B$, and $C$ is the ratio of their equivalent investments: $A : B : C = \frac{1}{9} : \frac{1}{16} : \frac{5}{12}$

To simplify this ratio, we multiply by the Least Common Multiple (LCM) of the denominators $9$, $16$, and $12$, which is $144$.

• $A$'s share in ratio = $\frac{1}{9} \times 144 = 16$
• $B$'s share in ratio = $\frac{1}{16} \times 144 = 9$
• $C$'s share in ratio = $\frac{5}{12} \times 144 = 60$

The simplified profit ratio is $16 : 9 : 60$.

5. Calculating $C$'s Profit:

• Total parts in the ratio = $16 + 9 + 60 = 85$ parts
• Total profit = ₹ $17,000$
• Value of $1$ part = $\frac{17,000}{85} = 200$

$C$'s profit share = $60 \text{ parts} \times 200 = \text{₹ } 12,000$

Therefore, $C$ will get a profit of ₹ $12,000$.