The process by which any given exponentiation of a binomial can be expanded is the binomial theorem. Let n be a positive integer, x, y ∈ C, then
(x + y)n = nC0 xn-0 y0 + nC1 xn-1 y1 + nC2 xn-2 y2 + ………. + nCr xn-r yr + …. nCn-1 xyn-1 + nCn x0 yn
(x + y)n = ∑r=0n nCr xn-r yr
Here, nC0, nC1, nC2 …… nCn are binomial coefficients.
Key features of the binomial theorem are as follows:
1) In the binomial expansion, there exists one extra term, which is more than that of the value of the index.
2) In the binomial theorem, the coefficients of binomial expressions are at the same distance from the beginning to the end.
3) an and bn are the 1st and final terms, respectively. x = y or x + y = n is valid if nCx = nCy.
4) C0 + C1 + C2 + ….. + Cn = 2n
5) C0 + C2 + C4 + ….. = C1 + C3 + C5 + ….. = 2n-1
6) C02 + C12 + C22 + ….. + Cn2 = 2nCn = (2n!) / n! n!
7) The general term of a binomial expansion is given by Tr+1 = nCr xn-r ar.
8) nCr + nCr-1 = n+1Cr
More on the properties of binomial coefficients are discussed below.
1) ∑r=0n nCr xn-r yr = (x + y)n
2) ∑r=0n nCr yr 1n-r = (1 + y)n
3) ∑r=0n nCr 1n-r 1r = (1 + 1)n = 2n
4) ∑r=0n nCr (- 1)r 1n-r = (1 + (- 1))n = 0
5) ∑r=0n n-1Cr-1 = ∑r=1n n-1Cr-1 1n-r (1)r-1 = (1 + 1)n-1 = 2n-1
6) ∑r=0n n+2Cr+2 = 2n+2 – n+2C0 – n+2C1
7) ∑r=0n n-1Cr-1 (- 1)r+1 = ∑ n-1Cr-1 (- 1)r-1 (- 1)2 = (1 + (- 1))n = 0
8) nCr = (n / r) (n-1Cr-1)
9) nCr / n = (n-1Cr-1) / r
10) (nCr) / (r + 1) = n+1Cr+1 / (n + 1)
11) (n+1Cr+1) / (r + 2) = n+2Cr+2 / (n + 2)
Example 1:
Solution:
Example 2: If (1 + x)n = C0 + C1 x + C2 x2 + …. + Cn xn, then
Solution:
(1 + x)n = C0 + C1x + C2x2 + ….. + Cnxn …..(i) and
(1 + (1 / x))n = C0 + C1 (1 / x) + C2 (1 / x)2 + ….. + Cn (1 / x)n ….(ii) If we multiply (i) and (ii), we get
C02 + C12 + C22 + ….. + Cn2 is the term independent of x, and hence, it is equal to the term independent of x in the product (1 + x)n (1 + (1 / x))n or in (1 / x)n (1 + x)2n or term containing xn in (1 + x)2n.
Clearly, the coefficient of xn in (1 + x)2n is Tn+1 and equal to 2nCn = (2n)! / n!n!
Example 3:
Solution:
Substituting n + 1 = N, we get
Trick: Put n = 1, then
Properties of Binomial Coefficients – Video Lesson
Frequently Asked Questions
Give the general formula of the binomial theorem.
The binomial theorem formula is (x + y)n = ∑r=0n nCr xn-r yr. Binomial coefficients are nC0, nC1, nC2 …… nCn.
How many terms are there in the binomial expansion of (x+a)n?
The total number of terms in the binomial expansion of (x+a)n is n+1 terms.
What do you mean by the constant term in the binomial theorem?
In the binomial expansion, the constant term is independent of the variables, and it is a numeric term.
Give two applications of the binomial theorem.
The binomial theorem is used to find the roots of equations in higher powers.
This theorem is also used in statistical and probability analysis.
Give the formula for the general term in the binomial theorem.
The general term of the binomial expansion of (x+y)n is Tr+1 = nCr xn-r yr.
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