Binomial Theorem 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. So, using this theorem even the coefficient of x20 can be found easily. The theorem plays a major role in determining the probabilities of events in the case of a random experiment. The binomial theorem expansion's exponent value can be a fraction or a negative number.
Read More: Polynomial Formula
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Key Terms: Binomial Expansion, Binomial Theorem, Pascal’s Triangle, Coefficients, Probability, Exponents, Power
What is Binomial Expression?
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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 and the distributive property must be used to multiply the binomials. For example, (1+x), (x+y), (x2+xy) and (2a+3b) are few binomial expressions.
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 and as the coefficients of ay.b(n-y) in (a+b)n, they are called binomial coefficients.
Pascal’s Triangle
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
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Binomial Expansion
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Using the Pascal triangle the binomial expansion can be written for (a+b)n. From the fifth row, the expansion of (a+b)4 can be written. And 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.
The binomial expansion consists of various terms that are:
General Term is given by
| Tr + 1 = nCran – rbr |
When n is even the total number of terms in expansion n + 1(odd). Then (n/2+1)th term is the middle term and is given by
| T(n/2 + 1) = nCn/2.an / 2.bn/2 |
When n is odd the total number of terms in expansion is n+1(even). Then ((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 |
and
| T((n+3)/2) = nCn-1 / 2.an-1 / 2.bn+1 / 2 |
- The total number of terms in an expansion of (a+b)n are n+1.
- Here n is the sum of powers of a and b.
Read More: Permutations and Combinations
Definition of Binomial Theorem
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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 ∑
And,
| (n/r) = nCr = n!/r!(n-r)! |
This is the binomial coefficient.
Binomial Theorem
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Formula for Binomial Theorem
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(a+b)n = nC0an b0 + nC1an-1b¹ +……..+ nCr an-r br +………+ nCn a0 bn
Here 1C0 = 1 and 1C1 = 1
So it can be inferred that
| (a+b)k = kC0 ak b0 + kC1ak-1 b1 +……..+ kCrak-r br +………+ kCk a0bk |
Read More: Bayes Theorem
Properties of Binomial Theorem
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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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Things to Remember
- The mathematical expression that consists of two terms including addition or subtraction operations is called the Binomial Expression.
- 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.
- Using the Pascal triangle the binomial expansion can be written for (a+b)n.
- Binomial Theorem states that If n is any positive integer, then,
(a+b)n = ∑(n/r)an-r. br where r = 0 to n for ∑
- (n/r) = nCr = n!/r!(n-r)! is a binomial coefficient.
- Formula for Binomial Theorem is given by,
(a+b)n = nC0 an b0 + nC1 an-1 b1 +……..+ nCr an-r br +………+ nCn a0 bn
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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