In this task you will be given a text passage from a trivia quiz. You should choose the category this question belongs to from these categories: History, Science, Social Science, Fine Arts, Literature. You can only choose one category, so if in doubt, please choose the closest match.

After allying with the khan of Chagatai and taking part in campaigns in Transoxania, this ruler would lose a "muddy battle" against Ilyas Khodja, despite fighting alongside brother-in-law Husayn of Balkh. Becoming the head of the Barlas upon the death of Tughluk, this man annihilated Tokhtamysh and the Golden Horde at the Battle of the Terek River before he overwhelmed Sultan Mehmud's forces in seizing the Delhi Sultanate. Prior to his death and ultimate succession by Shah Rukh while planning an invasion of China, he defeated the Turks at the Battle of Ankara and captured Bayezid I. FTP, name this Mongol military leader who ruled in Central Asia during the 14th century from his capital of Samarkand.
History

Laplaces rule may be used to compute a version of this quantity for its namesake tensor in order to prove the Alon-Tarsi conjecture for Latin squares of even order; that proof uses Cayleys version of this operator. The Cauchy-Binet rule generalizes it to commutative rings, while Leibnizs formula uses permutations of elements to compute it. Setting to zero the result of this operation on lambda I minus A allows us to calculate As eigenvalues. Cramers rule uses ratios of these to solve systems of linear equations. For ten points, name this quantity that is a d minus c b for a 2 by 2 matrix a b c d.
Science

If two variates obey this, their difference obeys a Skellam distribution. It has moment-generating function e to the quantity lambda times quantity e to the t minus one, where lambda is the only parameter on which this distribution depends. Bortkiewicz called it the "law of small numbers" after noting that it models rates of rare events like radioactive decays or horse-kick deaths. This function gives as lambda to the n times e to the minus lambda over n factorial as the probability of n successes from a large number of the trials named for its French namesake. FTP, name this distribution usually symbolized P.

Science