A and B are partner in a partnership firm sharing profit and losses in 3 : 1. C is admitted for $${\frac{1}{3}^{{\text{rd}}}}$$ share. New profit sharing ratio between A, B and C is

The correct answer is $\boxed{\text{B. }3:1:2}$.

The initial profit sharing ratio between A and B is 3:1. When C is admitted for a $\frac{1}{3}$ share, the new profit sharing ratio is calculated as follows:

• A’s new share = (3/4) x 3 = 3
• B’s new share = (1/4) x 1 = 1
• C’s new share = (1/3) = 2

Therefore, the new profit sharing ratio between A, B and C is 3:1:2.

Option A is incorrect because it does not take into account the new share that C is taking. Option C is incorrect because it does not reflect the fact that A and B are sharing the remaining 2 shares equally. Option D is incorrect because it does not reflect the fact that C is taking a $\frac{1}{3}$ share.