Binomial Theorem Formulas

Need any assistance on expanding Binomial Expressions and fed up doing the tedious task of calculating it manually. Thankfully you need not worry as we have curated the Binomial Theorem Formulas that makes your job simple. Look at the Binomial Theorem Cheat Sheet and get the expanded form effortlessly. You will feel the Binomial Formulae List given extremely useful while solving related problems.

Binomial Theorem Formulae List

Solving Binomial Theorem Related Problems can be really time-consuming and hectic. So, try solving the Binomial Expression Problems using the formulae listed and arrive at the solution easily. Basic & Advanced Binomial Theorem Formula Tables help you to cut through the hassle of doing lengthy calculations.

1. Binomial Theorem for positive Integral Index

(x + a)n = nC0xna0 + nC1 xn-1 a + nC2xn-2 a2 + …… + nCrxn-r ar+ …… + nCnxan

2. General term

(r + 1)th terms is called general term
Tr+1 = nCrxn-r ar

3. Deductions of Binomial Theorem

(i) (1 + x)n = nC0 + nC1x + nC2x2 + nC3x3 + …….. + nCrxr + ……… + nCnxn
which is the standard form of binomial expansion.
General term = (r + 1)th term:
Tr+1 = nCrxr = $$\frac{n(n-1)(n-2) \ldots .(n-r+1)}{r !}$$.xr

(ii) (1 – x)n = nC0nC1x + nC2x2nC3x3 + …….. + (-1)r nCrxr + ……… + (-1)n nCnxn General term = (r + 1)th term:
Tr+1 = (-1)r. nCrxr = (-1)r. $$\frac{n(n-1)(n-2) \ldots .(n-r+1)}{r !}$$.xr

4. Number of terms in the expansion of (x + y + z)n

n+2C2 = $$\frac{(n+1)(n+2)}{2}$$
Number of terms in the expansion of (x1 + x2 + x3 + …. + xk)n are n+k-1Ck-1 when x1, x2, x3 ………. xk all are different and can not be solved.

5. Middle term in the expansion of (x + a)n

• If n is even then middle term = $$\left(\frac{\mathrm{n}}{2}+1\right)^{\mathrm{th}}$$ term.
• If n is odd then middle terms are = $$\left(\frac{n+1}{2}\right)^{t h}$$ and $$\left(\frac{n+3}{2}\right)^{t^{\prime \prime}}$$ term. Binomial coefficient of middle term is the greatest Binomial coefficient.

6. To determine a particular term in the expansion

In the expansion of $$\left(x^{\alpha} \pm \frac{1}{x^{\beta}}\right)^{n}$$, if xm occurs in Tr+1, then r is given by
nα – r (α + β) = m ⇒ r = $$\frac{n \alpha-m}{\alpha+\beta}$$ and the term which is independent of x then nα – r (α + β) = 0 ⇒ r = $$\frac{n \alpha}{\alpha+\beta}$$

7. To find a term from the end in the expansion of (x + a)n

Tr(E) = Tn-r+2(B)

8. Binomial coefficients & their properties

In the expansion of (1 + x)n = C0 + C1x + C2x2 + …… + Crxr + ….+ Cnxn where C0 = 1, C1 = n, C2 = $$\frac{n(n-1)}{2 !}$$

• C0 + C1 + C2 + ……. + Cn = 2n
• C0 – C1 + C2 – C3 + …….. = 0
• C0 + C2 + ……… = C1 + C3 + …….. = 2n-1
• C02 + C12 + C22 + ……… + Cn2 = $$\frac{2 n !}{n ! n !}$$
• $$C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots .+\frac{C_{n}}{n+1}=\frac{2^{n+1}-1}{n+1}$$
• $$C_{0}-\frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4}+\ldots+\frac{(-1)^{n} \cdot C_{n}}{n+1}=\frac{1}{(n+1)}$$

9. Greatest term in the expansion of (x + a)n

(i) The term in the expansion of (x + a)n of greatest coefficient (ii) The greatest term
= $$\left\{\begin{array}{l}T_{p} \& T_{p+1}, \text { when } \frac{(n+1) a}{x+a}=p \in Z \\ T_{q+1}, \text { when } \frac{(n+1) a}{x+a} \notin Z \text { and } q<\frac{(n+1) a}{x+a}<q+1 \end{array}\right.$$
Note: Here take only positive values |x| and |a|

10. Binomial Theorem when “n” is any index

(1 + x)n = 1 + nx + $$\begin{array}{c} \frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{3 !} \mathrm{x}^{3}+\ldots \ldots . \\\ldots .+\frac{\mathrm{n}(\mathrm{n}-1) \ldots \ldots .(\mathrm{n}-\mathrm{r}+1)}{\mathrm{r} !} \mathrm{x}^{\mathrm{r}}+\ldots \ldots \infty \end{array}$$
General term: Tr+1 = $$\frac{n(n-1)(n-2) \ldots(n-r+1)}{r !} \cdot x^{r}$$

11. Some important expansions

• (1 -x)-1 = 1 + x + x2 + x3 + …….. + xr + …….. General term Tr+1 = xr
• (1 + x)-1 = 1 – x + x2 – x3 + …. (-x)r + …….. General term Tr+1 = (-x)r
• (1 – x)-2 = 1 + 2x + 3x2 + 4x3 + …….. + (r + 1) xr + ………. General term Tr+1 = (r + 1) xr
• (1 + x)-2 = 1 – 2x + 3x2 – 4x3 + …….. + (r + 1) (-x)r + ………. General term Tr+1 = (r + 1) (-x)r.

12. If ($$\sqrt{\mathrm{P}}$$ + Q)n = I + f where I and n are the integers, n being odd, and 0 ≤ f ≤ 1 then (I + f)f = kn, where P – Q2 = k > 0 and $$\sqrt{\mathrm{P}}$$ – Q < 1

13. Multinomial Expansion:

If n ∈ N then the general terms of the multinomial expansion (x1 + x2 + …….. + xk)n is 