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Binomial Theorem is a type of theorem that can be used for the algebraic expansion of binomial (a+b) for a positive integral exponent n. When the power of an expression increases, the calculation becomes difficult and lengthy.
- The binomial theorem exponent value can be a fraction or a negative number.
- The theorem plays a major role in determining the probabilities of events in the case of a random experiment.
- It was first introduced by Euclids around 400 B.C.
- Binomial Theorem is also known as binomial expansion.
- The solution can be obtained by multiplying the number of times based on the exponent value.
- So, using this theorem, even the coefficient of x20 can be found easily.
- The binomial theorem can be used to solve for expansion, which can be represented as:
(x + y)n = axuyc
- where n is a positive integer which is equal to u + c
- u and c are non-negative integer.
Key Terms: Binomial Expansion, Binomial Theorem, Pascal’s Triangle, Coefficients, Probability, Exponents, Power, Distributive Property, Exponent, Fraction, Binomial Coefficients
Binomial Theorem Statement
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Binomial Theorem is the mathematical expression that consists of two terms including addition or subtraction operations. The equal terms should be combined to add the binomials.
- The distributive property must be used to multiply the binomials.
- It can easily expand the polynomial function into suitable value.
- The coefficient of the expansion can be calculated using Pascal’s Triangle.
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The Binomial theorem for the expansion of (a+b)n is stated as,
(a+b)n = nC0an b0 + nC1an-1b¹ +……..+ nCr an-r br +………+ nCn a0 bn
Example of Binomial Theorem StatementExample 1: (1+x), (x+y), (x2+xy) and (2a+3b) are few binomial expressions. Example 2: When you purchase a medicine for illness then there are chances of getting cured with the medicine or not cured by the medicine. |
Binomial Theorem and Pascal Triangle
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Binomial Coefficients
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The coefficients in the binomial expansion of (a+b)n, n € N are called binomial coefficients. nC0, nC1, nC2 . . . . . . .nCn are some of the coefficients.
- Since nCr occurs as the coefficients of xx in (1+x)n where n€N.
- The coefficients of ay.b(n-y) in (a+b)n, they are called binomial coefficients.
- These coefficient values of nCr can be arranged in the form of a triangle and are called the Pascal triangle.
- The (k+1) row consists of values kC0, kC1, kC2, kC3,…….,kCk.
- It is used in the field of mathematics, and especially combinatorics.
Pascal’s Triangle
Binomial Expansion
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Using the Pascal triangle the binomial expansion can be written for (a+b)n. The total number of terms for an expansion is calculated to be n+1. The sum of exponents a and b is equal to n.
- The binomial expansion consists of various terms that are:
- General term for binomial expansion is as follows:
Tr + 1 = nCran – rbr
- Middle term of the expansion is given as:
T(n/2 + 1) = nCn/2.an / 2.bn/2
- Where n is even the total number of terms in expansion n + 1(odd) and (n/2+1)th term is the middle term
- When n is odd the total number of terms in expansion is n+1(even) and ((n+1)/2)th and ((n+3)/3)th terms are two middle terms.
- It is given by,
T((n+1)/2) = nCn-1 / 2.an+1 / 2.bn-1 / 2
T((n+3)/2) = nCn-1 / 2.an-1 / 2.bn+1 / 2
Example of Binomial ExpansionExample 1: From the fifth row, the expansion of (a+b)4 can be written. Example 2: From the sixth-row expansion of (a+b)5 can be written. So, we can write the expansion as (a+b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. |
| Class 11 Mathematics Related Concept | ||
|---|---|---|
| Real-Valued Function | Trigonometric Functions | Complex numbers and Quadratic Equations |
| Geometric Progression | Arithmetic Progression | Geometric Mean |
Binomial Theorem Coefficients
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The theorem used for expanding the binomial expression having infinite power is called the Binomial Theorem. It states that If n is any positive integer, then
(a+b)n = ∑(n/r)an-r.b∏
- where r = 0 to n for ∑
- The binomial coefficient is given as:
(n/r) = nCr = n!/r!(n-r)!
Binomial Theorem
Binomial Theorem Formula
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The binomial theorem formula for the expansion of (a+b)n is stated as,
(a+b)n = nC0an b0 + nC1an-1b¹ +……..+ nCr an-r br +………+ nCn a0 bn
- Here 1C0 = 1 and 1C1 = 1
- It can be inferred that
(a+b)k = kC0 ak b0 + kC1ak-1 b1 +……..+ kCrak-r br +………+ kCk a0bk
Example of Binomial Theorem FormulaExample: Determine the expansion of (x + 5)2 using the binomial theorem formula. Ans. As we know, (a+b)n = nC0an b0 + nC1an-1b¹ +……..+ nCr an-r br +………+ nCn a0 bn (x + 5)2 = x2 + 10x + 25 |
Properties of Binomial Theorem Coefficients
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For easy calculation the coefficients are given the values and certain formulas and represented as follows:
- C0 + C1 + C2 +…+ Cn = 2n
- C0–C1+C2 –…+(–1)nCn = 0
- C0 + C2 + C4 +…= C1 + C3 + C5 +…= 2n–1
- nCr = nCn–r
- r(nCr)=nn-1 Cr–1
- nCr/r+1 = (n+1)Cr+1/(n+1)
- nCr + nCr–1 = (n+1)Cr
- Where n ∈N, r ∈ W and r ≤ n
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Terms in Binomial Expansion
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Some important terms in binomial expansion are as follows:
General Term
General Term is used to represent all the values in an binomial expansion which is represented as
Tr + 1 = nCran – rbr
- Here r is the equivalent to one less than number of terms.
Example of General TermExample: Find the number of terms in (1 + 4x +7x2)50 Ans: (1 + 4x + 7x2)50 = [(1 + 7x)2]50 = (1 + 7x)100 The number of terms = (100 + 1) = 101 |
Middle Term
Middle term depends upon the value of n where n is a positive integer. If n is even the total number of terms in expansion is n + 1(odd) and (n/2+1)th term is the middle term
- When n is odd the total number of terms in expansion is n+1(even) and ((n+1)/2)th and ((n+3)/3)th terms are two middle terms.
Example of Middle TermExample: Find the middle term of (1 −4x + 4x2 )2n Ans: (1 − 4x + 4x2 )2n = [(1 − 2x)2]2n = (1 − 2x)4n Middle Term = [(4n/2) + 1] term |
Independent Term
The expression for independent term for expansion (axp + (b/xq)n can be represented as:
Tr+1 = nCr an-r br
- where r = (np/p+q) (integer)
Example of Independent TermExample: Find the independent term of x in (x+1/x)4 Ans: r = [4(1)/1+1] = 2 The independent term is 4C2 = 12 |
Numerically Greatest Term
Numerically Greatest Term is a terms which is calculated by first converting the term into binomial expansion form. In the second step integral value that is rounded to obtain the required value.
- The formula for numerically greatest term is given as:
[(n+1) |x|] / (1 + |x|)
Things to Remember
- Binomial Theorem states that If n is any positive integer, then, (a+b)n = ∑(n/r)an-r. br where r = 0 to n.
- Formula for Binomial Theorem is given by: (a+b)n = nC0 an b0 + nC1 an-1 b1 +……..+ nCr an-r br +………+ nCn a0 bn
- The mathematical expression that consists of two terms including addition or subtraction operations is called the Binomial Expression.
- Using the Pascal triangle the binomial expansion can be written for (a+b)n.
- An individual can practice Binomial Theorem Important Questions Mathematics
- Even students can practice NCERT Solutions For Class 11 Maths Chapter 8: Binomial Theorem
Previous Years’ Questions
- If some three consecutive in the binomial expansion of… [JEE Main – 2019]
- K(50C25), then K is equal to… [JEE Main – 2019]
- If the fractional part of the number… [JEE Main – 2019]
- For all x∈R, a0/a2 is equal to… [JEE Main – 2019]
- Then a - n is equal to… [BITSAT – 2017]
- The total number of terms in the expansion of… [KCET – 2017]
- sum of the coefficients of all the terms in this expansion, is… [JEE Main – 2016]
- If α and β be the coefficients of x4 and x2 respectively… [JEE Main – 2020]
- The expansion of (1+x)44 are equal, then x is equal to… [KCET – 2014]
- Coefficient of x11 in the expansion of… [JEE Advanced – 2014]
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Sample Questions
Ques. Expand (5x – 4)10 using Binomial Theorem. (2 Marks)
Ans. (5x – 4)10 = 10C0 (5x)10(–4)0 + 10C1 (5x)10-1 (–4)1 + 10C2 (5x)10-2 (–4)2 + 10C3 (5x)10-3 (–4)3 + 10C4 (5x)10-4 (–4)4 + 10C5 (5x)10-5 (–4)5 + 10C6 (5x)10-6 (–4)6 + 10C7 (5x)10-7 (–4)7 + 10C8 (5x)10-8(–4)8 + 10C9 (5x)10-9(–4)9 + 10C10 (5x)10-10(–4)10
Ques. Find the expansion of (x + y)6. (5 Marks)
Ans. (x + y)n = nC0xny0 + nC1xn-1 y1 + nC2xn-2 y2 + nC3 xn-3 y3 + ... + nCn−1x yn-1 + nCnx0 yn
(x + y)6 = 6C0x6 + 6C1x5 y + 6C2 x4y2 + 6C3x3y3 + 6C4x2y4 + 6C5xy5 + 6C6 y6
= ( 6! / [(6-0)!0!] ) x6 + ( 6! / [(6-1)!1!] ) x5 y + ( 6! / [(6-2)!2!] ) x4y2 + ( 6! / [(6-3)!3!] ) x3y3 + ( 6! / [(6-4)!4!] ) x2y4 + ( 6! / [(6-5)!5!] ) xy5 + ( 6! / [(6-6)!6!] ) y6
= x6 + 6x5 y + 15x4 y2 + 20x3 y3 + 15x2 y4 + 6x y5 + y6
Therefore, (x + y)6 = x6 + 6x5 y + 15x4 y2 + 20x3 y3 + 15x2 y4 + 6x y5 + y6
Ques. Find the third term in the expansion of (3 + y)6. (3 Marks)
Ans. As the expansion is of the form (a + x)n, so r th term
= an-r+1 xr-1 [{n(n–1) (n – 2) ... (n – r + 2)} ÷ (r – 1)!]
Here r = 3 and n = 6.
So 3rd term of (3 + y)6 = 3(6-3+1) . y(3-1) . [(6x5)/2]
=34. y2 . 15 = 1215 y2
Ques. Find the coefficient of p5 in the expansion of (p + 2)6. (5 Marks)
Ans. As expansion is of the form (x + a)n, so rth term
= xn-r+1 ar-1 [{n(n–1) (n – 2) ... (n – r + 2)} ÷ (r – 1)!].
So x5 will come when r = 2 and n = 6.
Hence we have to find the 2nd term of the expansion.
So r = 2 and n = 6.
So 2nd term of (p + 2)6 = p(6-2+1). 2(6-1)
= p5. 25. 6 = 192 p5
Hence coefficient of p5 is 192.
Ques. Find the coefficient of the independent term of x in expansion of (3x - (2/x2))15?. (2 Marks)
Ans.The general term of (3x - (2/x2)15 is given as Tr+1 = 15Cr (3x)15-r (-2/x2)r. It is independent of x if,
15 - r - 2r = 0 => r = 5
- T6 = 15C5(3)10(-2)5 =
- - 16C5 310.25
Ques. If the coefficient of (2r + 4)th and (r - 2)th terms in the expansion of (1+x)18 are equal then find the value of r. (5 Marks)
Ans. The general term of (1 + x)n is Tr+1 = Crxr
Hence coefficient of (2r + 4)th term will be
T2r+4 = T2r+3+1 = 18C2r+3
and coefficient or (r - 2)th term will be
Tr-2 = Tr-3+1 = 18Cr-3.
=> 18C2r+3 = 18Cr-3.
=> (2r + 3) + (r-3) = 18 (·.· nCr = nCK => r = k or r + k = n)
r = 6
Ques. Expand (2x + 3)? using Binomial Theorem. (3 Marks)
Ans. By comparing with the binomial formula, we get,
a = 2x, b =3 and n = 4.
Substitute the values in the binomial formula.
(2x + 3)4 = x4 + 4(2x)3(3) + [(4)(3)/2!] (2x)2 (3)2 + [(4)(3)(2)/4!] (2x) (3)3 + (3)4
= 16 x4 + 96x3 +216x2 + 216x + 81
Ques. How are binomials used in real life? (2 Marks)
Ans. Many cases of binomial expansion can be found, in actuality. For instance, if another medication is acquainted with fixing an infection, it either fixes the sickness (it's effective) or doesn't fix the illness (it's a disappointment). If you buy a lottery ticket, you're either going to win cash, or you're not.
Ques.How do you use Pascal's triangle? (2 Marks)
Ans. Perhaps the most intriguing Number Pattern is Pascal's Triangle (named after Blaise Pascal, a well-known French Mathematician and Philosopher). To construct the triangle, start with "1" at the top, then, at that point, keep setting numbers underneath it in a three-sided design. Each number is the numbers straight above it added together.
Ques. How do you identify a binomial? (2 Marks)
Ans. One can distinguish a random variable as being binomial if the following four requirements are met:
- There are a set number of trials (n).
- Each trial has two possible results: success or failure.
- The likelihood of success (call it p) is the same for each trial.
Ques. Which number is a binomial? (1 Mark)
Ans. In math, particularly in number theory, a binomial number is an integer that can be acquired by assessing a homogeneous polynomial containing two terms.
Ques. Where is binomial theorem used? (2 Marks)
Ans. The binomial theorem is used profoundly in Statistical and Probability Analyses. It is so helpful as our economy depends on Statistical and Probability Analysis. In more important math and calculation, the Binomial Theorem is used in attaining roots of equations in higher powers.
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