Which among the following numbers leaves remainders 1, 2 and 2 respectively when divided by 2, 3 and 7?

The best approach in these questions is to evaluate the options and arrive at the required answer by some trial and error. 
As we can see, the desired number has to be an odd number because 2 cannot divide it completely. Hence, options A and B get eliminated quickly. 
The required conditions get satisfied with option C.

The theoretical approach for such questions is long and not advisable when options are present.