**Given Information:**

A and B are partners in a firm sharing profits in the ratio of 3:2. C is admitted as a new partner on the condition that the profit sharing ratio between B and C will be 5:4.

**Solution:**

Let's assume the total profit of the firm is P.

**Step 1: Determine the profit sharing ratio between B and C:**

The profit sharing ratio between B and C is given as 5:4. This means that for every 5 units of profit B receives, C will receive 4 units.

Let's assume B's share of profit as 5x and C's share of profit as 4x.

**Step 2: Determine the total profit share of A, B, and C:**

A and B have a profit sharing ratio of 3:2. This means that for every 3 units of profit A receives, B receives 2 units.

Let's assume A's share of profit as 3y and B's share of profit as 2y.

**Step 3: Calculate the total profit share of each partner:**

Since the total profit of the firm is P, we can write the following equation:

3y + 2y + 5x + 4x = P

Combining like terms, we get:

5y + 9x = P ----(1)

**Step 4: Calculate the profit sharing ratio of A, B, and C:**

To find the individual profit sharing ratio of A, B, and C, we need to express their profit shares in terms of the total profit P.

The profit sharing ratio of A, B, and C can be calculated as follows:

A's profit sharing ratio = 3y/P
B's profit sharing ratio = 2y/P
C's profit sharing ratio = (5x + 4x)/P = 9x/P

**Step 5: Simplify the profit sharing ratios:**

To simplify the profit sharing ratios, we need to express them in their simplest form.

Dividing each ratio by the greatest common divisor (GCD) of their terms will give us the simplified profit sharing ratios.

Let's assume GCD(3y, 2y, 9x) = G

The simplified profit sharing ratios will be:

A's profit sharing ratio = (3y/G)/(P/G) = 3y/P
B's profit sharing ratio = (2y/G)/(P/G) = 2y/P
C's profit sharing ratio = (9x/G)/(P/G) = 9x/P

Therefore, the profit sharing ratio of A, B, and C is 3y:P, 2y:P, and 9x:P, respectively.

According to my calculations, answer is 15:10:8