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No, the dimensions of c, a and b can't be different.

According to the law of homogeneity of dimensions, the dimension of left side of equation (i.e. dimension of c) should be same as dimension of right side of equation (i.e. dimension of (a+b) ). 
Also, only quantity with same dimension can be added. So both a and b have same dimension which is the same as dimension of c.
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