For a positive integer n, expanding (a+b)n means choosing either a or b from each of n factors. A term containing r choices of b occurs nr times. (a+b)n=an+ n1an-1b+ n2an-2b2+. s+bn. Before like terms are collected, the expansion has n+1 binomial terms. The powers of a fall by one while those of b rise by one; their exponents total n. nr= n!r!(n-r)!, with r=0,1,,n and 0!=1. 102= 10 x 92 x 1=45 and 73= 7 x 6 x 53 x 2 x 1=35. The formula list supplies the positive-integer expansion and combination formula. On Paper 1, cancel factorial factors and retain exact arithmetic. On Paper 2, a calculator may evaluate the coefficient after the correct term has been chosen. For (a-b)n, use the whole second term -b; its power decides the sign.

For a positive integer \(n\), expanding \((a+b)^n\) means choosing either \(a\) or \(b\) from each of \(n\) factors. A term containing \(r\) choices of \(b\) occurs \(\binom nr\) times.

\((a+b)^n=a^n+\binom n1a^{n-1}b+\binom n2a^{n-2}b^2+\cdots+b^n\).

Before like terms are collected, the expansion has \(n+1\) binomial terms. The powers of \(a\) fall by one while those of \(b\) rise by one; their exponents total \(n\).

\(\binom nr=\dfrac{n!}{r!(n-r)!}\), with \(r=0,1,\ldots,n\) and \(0!=1\).

\(\binom{10}{2}=\dfrac{10\times9}{2\times1}=45\) and \(\binom73=\dfrac{7\times6\times5}{3\times2\times1}=35\).

The formula list supplies the positive-integer expansion and combination formula. On Paper 1, cancel factorial factors and retain exact arithmetic. On Paper 2, a calculator may evaluate the coefficient after the correct term has been chosen.

For \((a-b)^n\), use the whole second term \(-b\); its power decides the sign.

Positive integer expansion

Positive integer expansion