The Binomial Formula and Binomial Coefficients

Factorial n

If n = 1, 2 , 3, ..... factorial n or n factorial is defined as
3.1)            n! = 1 • 2• 3 • ..... • n
We also define zero factorial as
3.2)             0! = 1

Binomial Formula for Positive Integral n

If x and y are real numbers, then for all n Î N then

(x + y)ⁿ= C(n, 0)xⁿ + C(n, 1)x^n-1y + C(n, 2)x^n-2y² + … +C(n, r)x^n-ry^r + … + C(n, n)yⁿ

i.e. (x + y)ⁿ = SC(n, r)x^n-ry^r , r = 0 to n
3.3)             (x + y)ⁿ = xⁿ + nx^{n - 1}y + [n(n - 1)/2!].x^{n - 2}y² + [n(n - 1)(n - 2)/3!].x^{n - 3}y³ + ..... + yⁿ
This is called the binomial formula. It can be extended to other values of n and then is an infinite series.

Deductions from Binomial Theorem:

Replacing y by –y, we get:

(x – y)ⁿ = C(n, 0)xⁿ - C(n, 1)x^n-1y + C(n, 2)x^n-2y² - … +(-1)^rC(n, r)x^n-ry^r + … + (-1)ⁿC(n, n)yⁿ

(x - y)ⁿ = S(-1)^rC(n, r)x^n-ry^r , r = 0 to n

Replacing x by 1 and y by x, we get

(1 + x)ⁿ = C(n, 0) + C(n, 1)x + C(n, 2)x² + … + C(n, n)xⁿ

Replacing x by 1 and y by –x, we get

(1 - x)ⁿ = C(n, 0) - C(n, 1)x + C(n, 2)x² - … + (-1)ⁿC(n, n)xⁿ

Some Observations In A Binomial expansion:

(i)                           The expansion of (x + a)ⁿ contains (n + 1) terms.

(ii)                        Since C(n, r) = C(n, n-r), it follows that C(n, 0)=C(n, n), C(n, 1)=C(n, n-1)

(iii)                      Middle Terms In A Binomial Expansion: Since the expansion of (x + a)ⁿ contains n+1 terms, so

·        When n is even, then {(1/2)n + 1} th term is the middle term

·        When n is odd, then œ(n + 1)th and [œ(n + 1)+1]th terms are the two middle terms.

     (iv)       pth term from the end in (x + a)ⁿ = (n + 1 – p + 1)th  term from the beginning = (n – p + 2) th term from the beginning.

Binomial Coefficients

The result (3.3) can also be written
3.4

     
where the coefficinets, called binomial coefficients, are given by
3.5      

     

Properties of Binomial Coefficients

3.6             
This leads to Pascal's triangle

           

Multinomial Formula

where the sum, denoted by ∑, is taken over all nonnegative integers n₁, n₂, ....., n_p for which n₁ + n₂ + ... + n_p = n.