If x is a real number and k is a nonnegative integer, recall that the binomial coefficient (kx) is defined by the formula (kx)=k!x(x−1)(x−2)…(x−k+1).Compute the value of (20144028)(20141/2)⋅42014.
Solution — click to reveal
(20141/2)=2014!(1/2)(1/2−1)(1/2−2)⋯(1/2−2014+1)=2014!(1/2)(−1/2)(−3/2)⋯(−4025/2)=(2014!)22014(−1)(−3)⋯(−4025)=−(2014!)22014(1)(3)⋯(4025)⋅2⋅4⋅6⋅⋯⋅40262⋅4⋅6⋅⋯⋅4026=−(2014!)22014+2013(2013!)4026!So then (20144028)(20141/2)⋅42014=−(2014!)22014+2013(2013!)(20144028)4026!⋅42014=−(2014!)24027(2013!)(4028!)4026!⋅24028(2014!)(2014!)=−40271.