NEW DELHI—Instead of playing cricket with the kids in the alleyway outside, 4-year-old Sumit Jha sweats in his family's one-room apartment. A power cut has stilled the overhead fan. In the stifling heat, he traces and retraces the image of a goat.

In April, he enrolled in the nursery class of Shri Ram School, the most coveted private educational institution in India's capital. Its students include the grandchildren of India's most powerful figures—Prime Minister Manmohan Singh and Congress party President Sonia Gandhi.

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