Binomial Theorem Formulas

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 x^n-1 a + ⁿC₂x^n-2 a² + …… + ⁿC_rx^n-r a^r+ …… + ⁿC_nxaⁿ

2. General term

(r + 1)^th terms is called general term
T_r+1 = ⁿC_rx^n-r a^r

3. Deductions of Binomial Theorem

(i) (1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ⁿC₃x³ + …….. + ⁿC_rx^r + ……… + ⁿC_nxⁿ
which is the standard form of binomial expansion.
General term = (r + 1)^th term:
T_r+1 = ⁿC_rx^r = \(\frac{n(n-1)(n-2) \ldots .(n-r+1)}{r !}\).x^r

(ii) (1 – x)ⁿ = ⁿC₀ – ⁿC₁x + ⁿC₂x² – ⁿC₃x³ + …….. + (-1)^{r n}C_rx^r + ……… + (-1)^{n n}C_nxⁿ General term = (r + 1)^th term:
T_r+1 = (-1)^r. ⁿC_rx^r = (-1)^r. \(\frac{n(n-1)(n-2) \ldots .(n-r+1)}{r !}\).x^r

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 + …. + x_k)ⁿ are ^n+k-1C_k-1 when x₁, x₂, x₃ ………. x_k 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 x^m occurs in T_r+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

T_r(E) = T_n-r+2(B)

8. Binomial coefficients & their properties

In the expansion of (1 + x)ⁿ = C₀ + C₁x + C₂x² + …… + C_rx^r + ….+ C_nxⁿ where C₀ = 1, C₁ = n, C₂ = \(\frac{n(n-1)}{2 !}\)

• C₀ + C₁ + C₂ + ……. + C_n = 2ⁿ
• C₀ – C₁ + C₂ – C₃ + …….. = 0
• C₀ + C₂ + ……… = C₁ + C₃ + …….. = 2^n-1
• C₀² + C₁² + C₂² + ……… + C_n² = \(\frac{2 n !}{n ! n !}\)
• \(C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots .+\frac{C_{n}}{n+1}=\frac{2^{n+1}-1}{n+1}\)
• \(C_{0}-\frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4}+\ldots+\frac{(-1)^{n} \cdot C_{n}}{n+1}=\frac{1}{(n+1)}\)

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: T_r+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 + …….. + x^r + …….. General term T_r+1 = x^r
• (1 + x)^-1 = 1 – x + x² – x³ + …. (-x)^r + …….. General term T_r+1 = (-x)^r
• (1 – x)^-2 = 1 + 2x + 3x² + 4x³ + …….. + (r + 1) x^r + ………. General term T_r+1 = (r + 1) x^r
• (1 + x)^-2 = 1 – 2x + 3x² – 4x³ + …….. + (r + 1) (-x)^r + ………. General term T_r+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 + …….. + x_k)ⁿ is

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