BRAZIL

1. Choose the 1st letter any of 6 ways.
2. Choose the 2nd letter any of the 5 remaining ways. 
That's 6×5 or 30 ways to choose the first 2 letters.
3. Choose the 3rd letter any of the 4 remaining ways. 
That's 6×5×4 or 120 ways to choose the first 3 letters.
4. Choose the 4th letter any of the 3 remaining ways. 
That's 6×5×4×3 or 360 ways to choose the first 4 letters.
5. Choose the 5th letter either of the 2 remaining ways. 
That's 6×5×4×3×2 or 720 ways to choose the first 5 letters.
6. Choose the 6th letter the only 1 remaining way. 
That's 6×5×4×3×2×1 or 720 ways to choose the 6 letters. 

That's 6! = 6×5×4×3×2×1 = 720 ways

Then there are these two cases:

Case 1. Those with AI  (A on the left, I next to A on the right)
Case 2. Those with IA  (I on the left, A next to I on the right) 

Case 1. There are 5 things to arrange B, R, Z, L and AI   

1. Choose the 1st thing any of 5 ways.
2. Choose the 2nd thing any of the 4 remaining ways. 
That's 5×4 or 20 ways to choose the first 2 things.
3. Choose the 3rd thing any of the 3 remaining ways. 
That's 5×4×3 or 60 ways to choose the first 3 things.
4. Choose the 4th thing either of the 2 remaining ways. 
That's 5×4×3×2 or 120 ways to choose the first 4 things.
5. Choose the 5th thing the only 1 remaining way. 
That's 5×4×3×2×1 or 120 ways to choose the 5 things. 

That's 5! = 5×4×3×2×1 = 120 ways

Case 2. There are 5 things to arrange B, R, Z, L and IA

This is the same as case 1, or 120.

Total number of ways = 120+120 = 240 ways.

Edwin