✅ Binomial Theorem Formulas ⭐️⭐️⭐️⭐️⭐

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:

(ii) (1 – x)ⁿ = ⁿC₀ – ⁿC₁x + ⁿC₂x² – ⁿC₃x³ + …….. + (-1)^r nC_rx^r + ……… + (-1)^n nC_nxⁿ General term = (r + 1)^th term:

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

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

Binomial coefficient of middle term is the greatest Binomial coefficient.

6. To determine a particular term in the expansion

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

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

(i) The term in the expansion of (x + a)ⁿ of greatest coefficient

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.

13. Multinomial Expansion:

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

✅ 12th grade math formulas ⭐️⭐️⭐️⭐️⭐