**Binomial Theorem for Positive Integer**

If n is any positive integer, then

This is called binomial theorem.

Here, ^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2}, … , ^{n}n_{o} are called binomial coefficients and

^{n}C_{r} = n! / r!(n – r)! for 0 ≤ r ≤ n.

**Properties of Binomial Theorem for Positive Integer**

(i) Total number of terms in the expansion of (x + a)^{n} is (n + 1).

(ii) The sum of the indices of x and a in each term is n.

(iii) The above expansion is also true when x and a are complex numbers.

(iv) The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and

^{n}C_{r} = ^{n}C_{n – r}, r = 0,1,2,…,n.

(v) General term in the expansion of (x + c)^{n} is given by

T_{r + 1} = ^{n}C_{r}x^{n – r} a^{r}.

(vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

(vii)

(viii)

(ix) The coefficient of x^{r} in the expansion of (1+ x)^{n} is ^{n}C_{r}.

(x)

(xi) (a)

(b)

(xii) (a) If n is odd, then (x + a)^{n} + (x – a)^{n} and (x + a)^{n} – (x – a)^{n} both have the same number of terms equal to (n +1 / 2).

(b) If n is even, then (x + a)^{n} + (x – a)^{n} has (n +1 / 2) terms. and (x + a)^{n} – (x – a)^{n} has (n / 2) terms.

(xiii) In the binomial expansion of (x + a)^{n}, the r th term from the end is (n – r + 2)th term fromthe beginning.

(xiv) If n is a positive integer, then number of terms in (x + y + z)^{n} is (n + l)(n + 2) / 2.

**Middle term in the Expansion of (1 + x) ^{n}**

(i) It n is even, then in the expansion of (x + a)^{n}, the middle term is (n/2 + 1)^{th} terms.

(ii) If n is odd, then in the expansion of (x + a)^{n}, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.

**Greatest Coefficient**

(i) If n is even, then in (x + a)^{n}, the greatest coefficient is ^{n}C_{n / 2}

(ii) Ifn is odd, then in (x + a)^{n}, the greatest coefficient is ^{n}C_{n – 1 / 2} or ^{n}C_{n + 1 / 2} both being equal.

**Greatest Term**

In the expansion of (x + a)^{n}

(i) If n + 1 / x/a + 1 is an integer = p (say), then greatest term is T_{p} == T_{p + 1}.

(ii) If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then T_{m + 1}. is the greatest term.

**Important Results on Binomial Coefficients**

**Divisibility Problems**

From the expansion, (1+ x)^{n} = 1+ ^{n}C_{1}x + ^{n}C_{1}x^{2}+ … +^{n}C_{n}x^{n}

We can conclude that,

(i) (1+ x)^{n} – 1 = ^{n}C_{1}x + ^{n}C_{1}x^{2}+ … +^{n}C_{n}x^{n} is divisible by x i.e., it is multiple of x.

(1+ x)^{n} – 1 = M(x)

(ii)

(iii)

**Multinomial theorem**

For any n ∈ N,

(i)

(ii)

(iii) The general term in the above expansion is

(iv)The greatest coefficient in the expansion of (x_{1} + x_{2} + … + x_{m})^{n} iswhere q and r are the quotient and remainder respectively, when n is divided by m.

(v) Number of non-negative integral solutions of x_{1} + x_{2} + … + x_{n} = n is ^{n + r – 1}C_{r – 1}

**R-f Factor Relations**

Here, we are going to discuss problem involving (√A + B)sup>n = I + f, Where I and n are positive integers.

0 le; f le; 1, |A – B^{2}| = k and |√A – B| < 1

**Binomial Theorem for any Index**

If n is any rational number, then

(i) If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite.

(ii) General term in the expansion of (1 + x)^{n} is T_{r + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * xr}

(iii) Expansion of (x + a)^{n} for any rational index

(vii) (1 + x)^{– 1} = 1 – x + x^{2} – x^{3} + …∞

(viii) (1 – x)^{– 1} = 1 + x + x^{2} + x^{3} + …∞

(ix) (1 + x)^{– 2} = 1 – 2x + 3x^{2} – 4x^{3} + …∞

(x) (1 – x)^{– 2} = 1 + 2x + 3x^{2} – 4x^{3} + …∞

(xi) (1 + x)^{– 3} = 1 – 3x + 6x^{2} – …∞

(xii) (1 – x)^{– 3} = 1 + 3x + 6x^{2} – …∞

(xiii) (1 + x)^{n} = 1 + nx, if x^{2}, x^{3},… are all very small as compared to x.

**Important Results**

(i) Coefficient of x^{m} in the expansion of (ax^{p} + b / x^{q})^{n} is the coefficient of T_{r + l} where r = np – m / p + q

(ii) The term independent of x in the expansion of ax^{p} + b / x^{q})^{n} is the coefficient of T_{r + l}where r = np / p + q

(iii) If the coefficient of rth, (r + l)th and (r + 2)th term of (1 + x)^{n} are in AP, then n^{2} – (4r+1) n + 4r^{2} = 2

(iv) In the expansion of (x + a)^{n}

T_{r + 1} / T_{r} = n – r + 1 / r * a / x

(v) (a) The coefficient of x^{n – 1} in the expansion of

(x – l)(x – 2) ….(x – n) = – n (n + l) / 2

(b) The coefficient of x^{n – 1} in the expansion of

(x + l)(x + 2) ….(x + n) = n (n + l) / 2

(vi) If the coefficient of pth and qth terms in the expansion of (1 + x)^{n} are equal, then p + q = n + 2

(vii) If the coefficients of x^{r} and x^{r + 1} in the expansion of a + x / b)^{n} are equal, then

n = (r + 1)(ab + 1) – 1

(viii) The number of term in the expansion of (x_{1} + x_{2} + … + x_{r})_{n is n + r – 1C r – 1.}

(ix) If n is a positive integer and a_{1}, a_{2}, … , a_{m} ∈ C, then the coefficient of x^{r} in the expansion of (a_{1} + a_{2}x + a_{3}x^{2} +… + a_{m}x^{m – 1})^{n} is

(x) For |x| < 1,

(a) 1 + x + x^{2} + x^{3}+ … + ∞ = 1 / 1 – x

(b) 1 + 2x + 3x^{2} + … + ∞ = 1 / (1 – x)^{2}

(xi) Total number of terms in the expansion of (a + b + c + d)^{n} is (n + l)(n + 2)(n + 3) / 6.

**Important Points to be Remembered**

(i) If n is a positive integer, then (1 + x)^{n} contains (n +1) terms i.e., a finite number of terms. When n is general exponent, then the expansion of (1 + x)^{n} contains infinitely many terms.

(ii) When n is a positive integer, the expansion of (l + x)^{n} is valid for all values of x. If n is general exponent, the expansion of (i + x)^{n} is valid for the values of x satisfying the condition |x| < 1.