Binomial theorem

In elementary algebra, the binomial theorem is the identity that states that for any non-negative integer n,

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k},$

or, equivalently,

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}y^{k}x^{n-k},$

where

${n \choose k}={\frac {n!}{k!(n-k)!}}.$

One way to prove this identity is by mathematical induction.

The first several cases

$(x+y)^{0}=1\,$

$(x+y)^{1}=x+y\,$

$(x+y)^{2}=x^{2}+2xy+y^{2}\,$

$(x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}\,$

$(x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}\,$

$(x+y)^{5}=x^{5}+5x^{4}y+10x^{3}y^{2}+6x^{2}y^{3}+y^{5}\,$

$(x+y)^{6}=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6}\,$

Newton's binomial theorem

There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)ⁿ as an infinite series when n is not an integer or is not positive.

In popular culture

Sir W. S. Gilbert mentions the binomial theorem at least twice: Once in the Major General's Song in the Gilbert and Sullivan operetta The Pirates of Penzance ("About binomial theorem I'm teeming with a lot o' news -- / With many cheerful facts about the square of the hypotenuse"); and again in the poem "My Dream" in his "Bab Ballads," which says of a group of intelligent infants: "For as their nurses dandle them, / They crow binomial theorem, / With views (it seems absurd to us) / On differential calculus." (Ian Bradley, ed., The Complete Annotated Gilbert and Sullivan (Oxford: Oxford University Press, 1996), pp. 218-219.)