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Students found it relatively easy when they were asked to find the squares or cubes of numbers up to 50. For example, when they have to find (78)6 or (127)5, they face difficulty due to repeated multiplications. This is where the Binomial theorem makes our life easier. In the maths chapter 7 class 11, students will study how to expand (x + y)n or (x - y)n in an easier way, where n is an integer or a rational number. Binomial theorem class 11 notes also discuss how we get Pascal’s triangle from the expansion of (x + y)n.
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These class 11 maths chapter 7 NCERT notes contain many visual graphics to make learning easier for students. Also, the latest CBSE guidelines have been followed by Careers360 experts when they made these notes. Class 11 Math chapter 7 notes help students to find the entire chapter in one place during the revision after completing the textbook exercises. Students can also check the NCERT Exemplar Class 11 Maths Chapter 7 Binomial Theorem for extra practice purposes.
We know,
If we observe each expansion,
The total number of terms of each expansion is 1 more than the index of
In the expansion, the power of the first quantity is gradually decreasing that is by 1, and the power of the second quantity is gradually increasing by 1.
Suppose n is the index of
Now, arranging the coefficients of expansion with respect to their index.
If we observe the pattern, we can get the coefficient of the next index.
The pattern is given below.
Figure: 1
This diagram is known as Pascal's triangle.
If we apply Pascal's triangle rule, then the coefficients of
Similarly, the coefficient of
Now, we can apply the combination formula to find the coefficients.
Binomial coefficient (for a Positive integral index n)
where n and r are positive integers and
Figure 2 can be rewritten as:
Index Coefficients
Binomial Theorem:
Binomial theorem for any positive integer
Proof:
The proof is obtained by applying the principle of mathematical induction.
Let the given statement be
For
Thus,
Suppose
We shall prove that
Now,
[by actual multiplication]
(by using
Thus, it has been proved that
Properties of the Binomial coefficient
1.
2.
and
Hence, it is proven that both are equal.
3. If
4.
5. If n is a positive integer and
Here binomial coefficients are:
6. Total no. of terms of the given as
To find the middle terms of using the binomial theorem
The general equation of the binomial is given as:
There are two cases.
1. If n is odd.
The number of terms of
Here,
So there will be two middle terms.
That are
2. If n is even
The number of terms of
Here,
So there will be one middle term.
That is
To find the sum of the coefficient of the binomial terms, we have to put the value of
For example
Q. If the binomial expression is
Solution: We have to put the value of
So the Sum of the binomial coefficients is 1.
Some Properties of the Binomial coefficients
Put
Put
Some Particular expansions:
NCERT Class 12 Maths chapter 7 notes are useful in many ways.
NCERT Class 11 Notes Chapter Wise |
NCERT Class 11 Maths Chapter 7 Notes |
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