A, B and C entered into a partnership. After eight months, B and C left the business. Find the total profit at the end of year.

A. Annual profit of B is Rs. 400 more than that of A and Rs. 200 more than that of C.
B. Amount invested by C is 50% of total amount invested by A & B together.
C. Ratio of profit share of C and that of A and B together is 3 : 8.

The correct answer is None of the given statements can answer the question.

Let us analyze the information given in the question and the individual statements to see if we can find the total profit at the end of the year.

Information from the main question:
Let the investments of A, B, and C be $A$, $B$, and $C$ respectively.
A, B, and C entered into a partnership.
A remained in the business for the entire year (12 months).
B and C left the business after 8 months, meaning they were invested for 8 months each.
Therefore, the ratio of their profit shares is given by:
$P_A:P_B:P_C=(A\times 12):(B\times 8):(C\times 8)=3A:2B:2C$

Let us evaluate each statement to see if we can determine the total profit ($P=P_A+P_B+P_C$):

Statement A:
Annual profit of B is Rs. 400 more than that of A and Rs. 200 more than that of C.
$P_B=P_A+400$
$P_B=P_C+200$
This gives us relations between the individual profits: $P_A=P_B-400$ and $P_C=P_B-200$.
However, we do not know the actual ratio of profits because the investment values are unknown. Thus, Statement A alone is not sufficient.

Statement B:
Amount invested by C is 50% of the total amount invested by A and B together.
$C=0.5(A+B)$
This statement only gives a relationship between the investments, but no monetary values for profits or investments are provided. Thus, Statement B alone is not sufficient.

Statement C:
Ratio of profit share of C and that of A and B together is 3 : 8.
$\frac{P_C}{P_A+P_B}=\frac{3}{8}$
This only gives a ratio and no absolute monetary values. Thus, Statement C alone is not sufficient.

Combining the Statements:
Let us try to combine the statements to see if a solution is possible.
From Statement A, we have:
$P_A=P_B-400$
$P_C=P_B-200$
If we combine this with Statement C:
$\frac{P_B-200}{(P_B-400)+P_B}=\frac{3}{8}$
$\frac{P_B-200}{2P_B-400}=\frac{3}{8}$
Cross-multiplying, we get:
$8(P_B-200)=3(2P_B-400)$
$8P_B-1600=6P_B-1200$
$2P_B=400$
$P_B=200$
Now we can compute individual profits:
$P_A=P_B-400=200-400=-200$
Since profit cannot be negative, this system yields an mathematically invalid result ($P_A=-200$), showing that Statements A and C are contradictory or incompatible under ordinary profit assumptions.

Even if we attempt to use Statement B, it relates investments:
$C=0.5(A+B)\Rightarrow 2C=A+B$
Using the profit relations from the time periods:
$P_A=3A·k$, $P_B=2B·k$, $P_C=2C·k$ (where $k$ is a constant of proportionality)
From Statement B:
$P_C=2C·k=(A+B)·k=\frac{P_A}{3}+\frac{P_B}{2}$
Comparing this relationship with Statement C, which states:
$\frac{P_C}{P_A+P_B}=\frac{3}{8}\Rightarrow P_C=\frac{3}{8}(P_A+P_B)$
Neither of these sets of relations allows us to determine a unique, positive total profit because they only define ratios, and combining them with Statement A leads to negative/non-real profit values.

Hence, even by combining the statements, we cannot find the total profit at the end of the year.