Basic Concepts of Binomial Theorem:
We have always done calculation to find the squares and cubes of binomials like (c+d) or (c-d), but what about higher powers like (115)^{7} or (67)^{6}, in these cases calculations become difficult because of repeated multiplications. This difficulty was overcome by a theorem called Binomial Theorem.
A hypothesis that indicates the expansion of a binomial of the form (x + y) to the exponent n as the total of n + 1 terms of which the general term comprises of a result of x and y with x^{(n-k) }and y^{k} and a coefficient comprising of n! separated by (n - k)!k! where k takes on value from 0 to n.
Binomial Theorem for Positive Integral Indices:
(x + y)^{0} = 1
(x + y)^{1} = x + y
(x + y)^{2} = x^{2} + 2xy + y^{2}
(x+ y)^{3} = a^{3} + 3x^{2}y + 3xy^{2} + y^{3}
Key Observations:
1) Total number of terms in the expansion is one more than the index i.e. in the expansion of (a + b)^{1 }, where the index is 1, there are two terms a and b.
2) Power of 'a' decreases by 1 and power of 'b' increases by 1 with every successive terms
3) In expansion, the sum of the indices of a and b is the same and is equal to the index of a+b.
Index Coefficients
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
This structure shown above is called as Pascal's Triangle. The coefficients of the expansions are arranged in an array, which in known as Pascal's Triangle.
When n is a positive integer, then binomial theorem is :
(x+y)^{n} = ^{n}c_{0}.x^{n} + ^{n}c_{1}x^{n-1}y + ^{n}c_{2}x^{n-2}y^{2} + ^{n}c_{3}.x^{n-3}y^{3} + ....... + ^{n}c_{r}x^{n-r }y^{r} + .... + ^{n}c_{n}.y^{n}
General Term in a binomial expansion:
In the binomial expansion of (x+y)^{n} , general term is denoted by T_{r + 1} and it is
T_{r + 1} = ^{n}c_{r}.x^{n - r}.y^{r}
Combinations or groups formula:
^{n}c_{r} = n!/[( n - r ) !].[r!]
Middle term in a binomial expansion:
In the binomial expansion of (x+y)^{n}, middle term is T_{( n/2 + 1)} if n is even, and T_{(n + 1)/2} and T_{( n + 3)/2} , if n is odd.
Binomial Coefficients in the binomial expansion (x+y)^{n}
^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2}, ^{n}C_{3},..... ^{n}C_{r}... ^{n}C_{n} are called Binomial Coefficients.
Binomial Coefficient of y^{m} in (ay^{s} + b / y^{t} )
The value of r of the term which contains the coefficient of y^{m} is (ns - m )/( s + t)
Independent Term of y in (ay^{s} + b / y^{t} )
The value of r of the term which does not contain y is ( ns ) / (s+ t)
Greatest Binomial Coefficients:
In the binomial expansion of (x + y)^{n} , the greatest binomial coefficient is ^{n}c_{(n+1)/2} , ^{n}c_{( n + 3 )/2} and ^{n}c_{( n/2 + 1)} . In first formula, n is an odd integer and in second formula n is an even integer.
Numerically Greatest term in the binomial expansion: (1 + y)^{n}
In the binomial expansion of (1 + x)^{n}, the numerically greatest term is found by the following method:
If [( n + 1 ) | y | ] / _{[| y | + 1}] = T + f,
Where T is an integer and f is a positive fraction, then
( T + 1)^{ th} term is the numerically greatest fraction.
And if [( n + 1 ) | y | ] /_{[| y | + 1}] = T,
Where T is an integer, then
T^{th} term and (T + 1 )^{th} terms are the two numerically greatest terms.
In the binomial expansion of (x+y)^{n} :
1. Sum of the binomial coefficients is 2^{n}
^{n}c_{0} + ^{n}c_{1} + ^{n}c_{2} + ............. + ^{n}c_{n} = 2^{n}
2. Sum of the odd binomial coefficients is 2^{n - 1}
c_{1} + c_{3} + c_{5} + ............. = 2^{n - 1}
3. Sum of the even binomial coefficients is 2^{n - 1}
c_{0} + c_{2} + c_{4} +........... = 2^{n - 1}
Number of terms in the expansion of :
1. ( a + b)^{n} is n + 1
2. ( a + b + c)^{ n} is ^{[( n + 1 ) ( n + 2 )]}/_{2}
3. ( a + b + c + d) ^{n} = ^{[ ( n + 1)(n + 2 ) ( n + 3 )]}/_{ 1. 2.3}
Importance of Binomial Theorem:
IP address Distribution: Application of Binomial Theorem comes extremely helpful in the circulation of IP, by appropriating bits to cover all the host, where numbers have are more prominent than the aggregate number of given IP address of the settled host. This procedure in known as subnetting.
National monetary expectation: Binomial Theorem is by and large utilized by Economists to foresee the conduct of the economy in the coming future years by tallying out probabilities, reliant on different circulated factors. It is helpful in finding the event of benefit and misfortune which is exceptionally beneficial for. creating economy.
Design: Engineers utilizes this technique to figure the sizes of the activities to convey exact assessments of cost and time required for developments. For contractual workers, it is a valuable apparatus to guarantee the costing activities is sufficiently skilled in conveying benefits.
Climate estimating: the Binomial hypothesis is utilized as a part of figure administrations, significantly debacle conjecture relies on the utilization of binomial hypotheses.
The binomial hypothesis is exceptionally valuable in the higher numerical operations and furthermore utilized by researchers in logical research to fathom outlandish conditions. In a hefty portion of the aggressive exams, it is utilized to ascertain the positions of the showed up competitors.
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