Quant · Algebra

Binomial Theorem

2 sub-topics · 18 lessons · 60 questions · 4 PYQs

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Binomial Coefficients and Terms11 lessons · 38 questions · 4 PYQs

Binomial Applications7 lessons · 22 questions

Formula sheet

Remainders by binomial expansion
$(km\pm c)^n\equiv(\pm c)^n\pmod{m}$
Binomial approximation
$(1\pm h)^n\approx1\pm nh+\binom{n}{2}h^2$
Ratio of consecutive coefficients
$\frac{\binom{n}{r+1}}{\binom{n}{r}}=\frac{n-r}{r+1}$
Partial alternating sum
$\sum_{r=0}^{j}(-1)^r\binom{n}{r}=(-1)^j\binom{n-1}{j}$
Equal binomial coefficients
$\binom{n}{a}=\binom{n}{b},\ a\ne b\ \Rightarrow\ a+b=n$
Hockey-stick identity
$\binom{k}{0}+\binom{k+1}{1}+\cdots+\binom{k+m}{m}=\binom{k+m+1}{m}$
Divisibility pattern from the first two terms
$(1+k)^n=1+nk+\binom{n}{2}k^2+\cdots\ \Rightarrow\ k^2\mid(1+k)^n-nk-1$
Binomial expansion and general term
$(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^{r},\qquad T_{r+1}=\binom{n}{r}a^{n-r}b^{r}$