Binomial Theorem

Obvious is the most dangerous word in mathematics. — E. T. Bell

7.1 Binomial Theorem (Positive Integral Index)

For a natural number $n$,

$(a+b)^n=\sum_{r=0}^{n}{}^nC_r\, a^{n-r}b^r$

that is

$(a+b)^n={}^nC_0 a^n+{}^nC_1 a^{n-1}b+\cdots+{}^nC_n b^n$

There are $n+1$ terms. The general term (term involving $b^r$) is

$T_{r+1}={}^nC_r\, a^{n-r}b^r$

7.2 Special Cases

$(1+x)^n=\sum_{r=0}^{n}{}^nC_r x^r$

$(x+y)^n=(y+x)^n$

The coefficients ${}^nC_0,{}^nC_1,\ldots,{}^nC_n$ are the entries of the $n$th row of Pascal’s triangle.

7.3 Simple Applications

Expand $(2x-3y)^4$, find a particular term from $T_{r+1}$, or evaluate numerically by substituting $a$ and $b$.

Chapter Summary

• $(a+b)^n=\sum{}^nC_r a^{n-r}b^r$
• $T_{r+1}={}^nC_r a^{n-r}b^r$
• $n+1$ terms; coefficients from ${}^nC_r$

Exercises (NCERT)

• Exercise 7.1 — expansions and general term
• Miscellaneous Exercise