A and B are partners sharing profits and losses in proportion 2 : 1. They admits new partner C hom they give $${\frac{1}{5}^{{\text{th}}}}$$ share in profits. The new profit sharing ratio will be:

The correct answer is $\boxed{\text{(C)}}$.

The initial profit sharing ratio of A and B is 2:1. This means that A gets 2 parts of the profit and B gets 1 part of the profit. When a new partner C is admitted, the new profit sharing ratio is determined by the following formula:

$$\text{New profit sharing ratio} = \frac{\text{Old profit sharing ratio} \times \text{Number of old partners}}{\text{Number of old partners} + \text{Number of new partners}}$$

In this case, the new profit sharing ratio is:

$$\text{New profit sharing ratio} = \frac{2 \times 2}{2 + 1} = \frac{4}{3}$$

This means that A gets 4 parts of the profit, B gets 2 parts of the profit, and C gets 1 part of the profit.

Option (A) is incorrect because it gives A 8 parts of the profit, which is more than the total number of parts (3).

Option (B) is incorrect because it gives A 3 parts of the profit, which is less than the amount that A gets in the initial profit sharing ratio (2 parts).

Option (D) is incorrect because it gives A 4 parts of the profit, which is the same amount that A gets in the initial profit sharing ratio.