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D e a r S t u d e n t, S o l u t i o n) T h e c o e f f . o f x^{3} i n t h e e x p a n s i o n o f {(1 - x + x^{2})}^{5} i s {}^{5}C_{0} - {}^{5}C_{1} (1 - x) . x + {}^{5}C_{2} (x {(1 - x))}^{2} - {}^{5}C_{3} (x {(1 - x))}^{3} + {}^{5}C_{4} (x {(1 - x))}^{4} - {}^{5}C_{5} (x {(1 - x))}^{5} c o e f f o f x^{3} i n t h e e x p a n s i o n i s : {}^{5}C_{2} (x {(1 - x))}^{2} - {}^{5}C_{3} (x {(1 - x))}^{3} = {}^{5}C_{2} . x^{2} . {(1 - x)}^{2} - {}^{5}C_{3} . x^{3} . {(1 - x)}^{3} = {}^{5}C_{2} . x^{2} . (1 + x^{2} - 2 x) - {}^{5}C_{3} . x^{3} . (1 - x^{3} - 3 x + 3 x^{2}) = - 2 . {}^{5}C_{2} + (- {}^{5}C_{3}) = - 2 \times \frac{5!}{2! \times 3!} - \frac{5!}{3! \times 2!} = - 3 \times \frac{5!}{3! \times 2!} = - 3 \times \frac{5.4.3!}{3! . 2.1} = - 30 R e g a r d s!

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