Is there any possible way to become software engineer without studying science?

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Its not possible.
The Foremost thing that you should keep in your mind is that, you have to take PCM(Physics, Chemistry, Maths) for pursuing any branch of engineering that is you have to pursue science stream for gettting into any branch of engineering(CS, IT, ME, EEE, EC, CIVIL).
Now coming to the other part, since you aspire of becoming a software engineer, branches like CS and IT are suitable for you.

In 12th, you can take any subjects along with PCM to get into the engineering branch. The other subjects doesn't matter much. But if you take subjects like Informatics Practices(IP) and Computer science(CS) in 11th then they will help you to build a platform for your future engineering.
Other subjects like Eco, or Physical education will also not make much difference.
In engineering you are taught from your basics, you need not to worry about that.

So, don't worry about the subjects, the only compulsory subjects are PCM i.e. science stream.