by 
James D
ANSWERS: 5
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I guess he saw in her a willingness to learn and unrelenting determination and decided she was ready. Afterall she did eventually single handedly kill the entire "Deadly viper assassination squad" and many others.
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He was wise enough to see a true student, one that would use her training with widsom. He had concerns about the others.
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Because he basically knew she was a bad ass!
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When asked the same question by Bill , Bill said- (Pai-Mei taught you the five-point exploding heart technique? ) Beatrix Kiddo replyed ( "of course he did") Which meant she was an exceptional student! Kill Bill volume II
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Because he fell in love with her. In a realistic sense off course but he must have felt something for her. This love that she awoke in him summarises all of her qualities as a student and as a person as opposed to other students. For Pai-Mei to love someone that someone has to be special in many ways. So he granted her the wisdom.
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