# Binomial theorem

${\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}}$
The binomial coefficient ${\displaystyle {\tbinom {n}{b}}}$ appears as the bt entry in the nt raw o Pascal's triangle (coontin stairts at 0). Ilk entry is the sum o the twa abuin it.

In elementar algebra, the binomial theorem (or binomial expansion) descrives the algebraic expansion o pouers o a binomial. Accordin tae the theorem, it is possible tae expand the polynomial (x + y)n intae a sum involvin terms o the form a xbyc, whaur the exponents b an c are nonnegative integers wi b + c = n, an the coefficient a o ilk term is a speceefic positive integer depending on n and b. For ensaumple (for n = 4),

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