<prompt> Mgawo wa x50 in1 x1000 2x 1 x999 3x2 1 x998 nukta 1001x1000inaweza kuonyeshwa katika fomu binomnk. Tafuta thamani ndogo kabisa ya n k.  <response>: HebuS 1 x1000 2x 1 x999 vidoti 1000x999 1 x 1001x1000KishaanzaalignxS x 1 x1000 2x2 1 x999 vidoti 1000x1000 1 x 1001x1001, 1 xS 1 x1001 2x 1 x1000 nukta 1000x999 1 x2 1001x1000 1 x.endalignTunapata kuondoa milinganyo hiiS 1 x1001 x1 x1000 vitone x999 1 x2 x1000 1 x - 1001x1001.KishaanzaalignxS x1 x1001 x2 1 x1000 vitone x1000 1 x2 x1001 1 x - 1001x1002, 1 xS 1 x1002 x 1 x1001 vitone x999 1 x3 x1000 1 x2 - 1001x1001 1 x.endalignTunapata kuondoa milinganyo hiiS 1 x1002 - 1002x1001 1 x 1001x1002.Kwa Nadharia ya Binomia, basi mgawo wa x50 binom100250. Jibu la mwisho ni 1002 50 1052. 