A, B and C are partners in a firm sharing profits in the ratio $$\frac{2}{5}:\,\frac{2}{5}:\,\frac{1}{5}.$$ C retires from the firm and his share is bought by A and B in equal ratio. New profit sharing ratio will be.

The correct answer is $\boxed{\text{A}}$.

Initially, A, B, and C share profits in the ratio $\frac{2}{5}:\,\frac{2}{5}:\,\frac{1}{5}$. This means that A’s share is $\frac{2}{5}$ of the total profit, B’s share is $\frac{2}{5}$ of the total profit, and C’s share is $\frac{1}{5}$ of the total profit.

When C retires, his share is bought by A and B in equal ratio. This means that A and B each acquire $\frac{1}{2}$ of C’s share. Therefore, A’s new share is $\frac{2}{5} + \frac{1}{2} = \frac{3}{5}$, and B’s new share is $\frac{2}{5} + \frac{1}{2} = \frac{3}{5}$.

Therefore, the new profit sharing ratio is $\text{A} = \frac{3}{5}:{\text{B}} = \frac{3}{5}$.

Option B is incorrect because it does not take into account the fact that C’s share is bought by A and B in equal ratio. Option C is incorrect because it does not take into account the fact that A’s share is $\frac{2}{5}$ of the total profit, B’s share is $\frac{2}{5}$ of the total profit, and C’s share is $\frac{1}{5}$ of the total profit. Option D is incorrect because it does not take into account the fact that A and B each acquire $\frac{1}{2}$ of C’s share.