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in Binomial theorem by (75 points)
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The value of \( \sum_{ k =0}^{14}(-1)^{ k }(15- k )^{3}{ }^{17} C _{ k } \) is equal to

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 \(\sum\limits^{14}_{k=0}\)(-1)k(15 - k)3 17Ck

= 153 17C0 - 143 17C1 + 133 17C2 - 123 17C3 + ...+33 17C12 - 23 17C13 + 13 17C14

 = 153 17C0 - 143 17C1 + 133 17C2 + (13 - 123) 17C3 + (113 - 23)17C4 

+ (33 - 103)17C5 + (93- 43)17C6  + (53 - 83)17C7 + (73 - 63) 17C8

(\(\because \) nCr - nCn-r)

 = 153 - 17 x 143\(\frac{17.16}2\times13^3+\frac{17.16.15}6(1-1728)+\frac{17.16.15.14}{24}\) \(\times(1331-8)+\frac{17.16.15.14.13}{120}\)

\(\times(27-1000)\)+\(\frac{17.16.15.14.13.12}{720}(729-64)\) + \(\frac{17.16.15.14.13.12.11}{5040}\times(125-512)\) \(+\frac{17.16.15.14.13.12.11.10}{40320}(343-216)\)

 = 3375 - 46648 + 298792 - 1174360 +3148740 -6020924 +8230040 -7526376 + 3087370

 = 14768317 - 14768308

 = 9

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