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India has several gender gap problems but it is not like Saudi Arabia, right? You are making it sound like it is. As far as I know, at the state level, it does not put obstacles in order to prevent women from being able to work. On the contrary, it has invested a lot to reverse the gap with various campaigns and attempts to increasing women's literacy rates and its role in the workforce. Today they already represent about 30% of the workforce. The Global Gender Gap report puts India on the same range as China.
As for education, 54.8% of women over 25 had secondary education as of 2010 in China. In India it is about 41%. They are getting closer. I know China has a better education system but not all Chinese are highly educated, the majority is not. Just like in India, it varies greatly from region to region. What they both have is an minority elite of brains, some of the brightest in the world even, only China does a better job of keeping them at home (often through dubious practices), while in India they tend to head for Silicon Valley.
Keep in mind as well that higher education also doesn't matter much when you spend your day assembling telephones or sewing clothes. And it is these kinds of industries and services that China and India cater to, innovative homegrown technology is still a niche. The main competitive advantage that China has over India is in fact just as you said, infrastructure. India still has a lot to do but that's exactly why it has room to grow.
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