Let's break down the problem step by step. 

Initial Positions: 

A is 10th from the left. 

B is 9th from the right. 

After Interchange: 

A takes B's position and becomes 15th from the left. 

Determine Total Boys in the Row 

Since A moves to the position where B was, the 15^th position from the left must be B's original position. This means B's original position is the 15th position from the left. 

To find the total number of boys in the row, we need to determine the position of B from the left and right: 

• From the right, B was originally 9^th. 
• From the left, B's position is now known to be 15^th. 

The total number of boys in the row can be calculated using the formula: 

• Total number of boys = Position from the left + Position from the right-1 

Total number of boys = 15+9-1 = 23 

So, there are 23 boys in the row.