# ✅ Binomial Theorem Formulas ⭐️⭐️⭐️⭐️⭐

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1. Binomial Theorem for positive Integral Index

(x + a)n = nC0xnanC1 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:

(ii) (1 – x)n = nC0 – nC1x + nC2x2 – nC3x3 + …….. + (-1)r nCrxr + ……… + (-1)n nCnxn 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 + …. + 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

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

Tr(E) = Tn-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 + …….. + 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.

13. Multinomial Expansion:

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