Case 1. The number has 3 digits and the first digit is 1.
There are only three, 152, 153, and 154

Case 2: The number has 3 digits and the first digit is not 1.
(They will all be greater than 150.)

We can choose the first digit 4 ways, (2,3,4,or 5).  
We can choose the second digit 4 ways.
We can choose the third digit 3 ways.

That's 4∙4∙3 = 48

Case 3: The number has 4 digits. 
(They will all be greater than 150.)

We can choose the first digit 5 ways.
We can choose the second digit 4 ways.
We can choose the third digit 3 ways.
We can choose the fourth digit 2 ways.  

That's 5∙4∙3∙2 = 120

Case 4: The number has 5 digits. 
(They will all be greater than 150.)

We can choose the first digit 5 ways.
We can choose the second digit 4 ways.
We can choose the third digit 3 ways.
We can choose the fourth digit 2 ways.  
We can choose the fifth digit 1 way.

That's 5∙4∙3∙2∙1 = 120

Total for all 4 cases: 3+48+120+120 = 291

Edwin