\(\sum\limits^{14}_{k=0}\)(-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 + \(\frac{17.16}3\times13^3+\frac{17.16.15}6(1-1728)+\frac{17.16.15.14}{24}\) \(\times(1331.8)+\frac{17.16.15.14.13}{120}\)
\(\times(27-100)\)+\(\frac{17.16.15.14.13.12}{720}(729-64)\) + \(\frac{17.16.15.14.13.12.11}{5040}\times(125-512)\) \(+\frac{17.16.15.14.13.12.11.10}{40320}(343-216)\)
= 3375 - 46648 + 298792 - 1174360 +3148740 -6020924 +8230040 -7526346 + 3087370
= 14768317 - 14768308
= 9