Mathematics Lab Activity-13 Class XI

Mathematics Laboratory Activities on Binomial Theorem for class XI students Non-Medical. These activities are according to the CBSE syllabus

Chapter - 7 Binomial Theorem
Activity - 13

Objective

To construct a Pascal’s triangle and to write binomial expansion for a given positive integral exponent.

Material Required

Drawing board, adhesive, white paper, matchsticks.

Theory

The expansion of a binomial for any positive integral n is given by Binomial theorem, according to which,

$(a+b)^{n}=^{n}C_{0}\:a^{n}b^{0}+^{n}C_{1}\:a^{n-1}b^{1}+^{n}C_{2}\:a^{n-2}b^{2}.....^{n}C_{r}\:a^{n-r}b^{r}....^{n}C_{n}\:a^{0}b^{n}$

The coefficients of the expansion are arranged in an array. This array is called Pascal’s triangle.

Procedure

1. Take a drawing board and paste a white paper on it.

2. Take some matchsticks and arrange them as triangles as shown in fig 13.1

3. Write the numbers as follows

First row 1

Second row 1, 1

Third row 1, 2, 1

Fourth row 1, 3, 3, 1

Fifth row 1, 4, 6, 4, 1

Sixth row 1, 5, 10, 10, 5, 1 and so on as shown in the figure 13.1

Fingure 13.1

4. This array of numbers in the triangular form is called a Pascal’s triangle, after the name of the French Mathematician, Blaise Pascal.

5. The number in the first row ‘1’ give the coefficient of the term of the binomial expansion of (a + b)⁰.

6. The number in the second row ‘1, 1’ give the coefficient of the term of the binomial expansion of (a + b)¹.

7. The number in the third row ‘1, 2, 1’ give the coefficient of the term of the binomial expansion of (a + b)².

8. The number in the fourth row ‘1, 3, 3, 1’ give the coefficient of the term of the binomial expansion of (a + b)³ and so on.

9. Students can extend their knowledge by writing more rows on the pascal triangle as shown in the figure 13.2

Figure 13.2

On the basis of this activity students can also made some useful and beautiful patterns as shown below in figure 13.3

Figure 13.3

Observations

From the Pascal’s triangle we find that

1. Numbers in the fifth row are 1, 4, 6, 4, 1, these are the coefficients of the binomial expansion of (a + b)⁴

2. Numbers in the sixth row are 1, 5, 10, 10, 5, 1, these are the coefficients of the binomial expansion of (a + b)⁵.

3. Numbers in the seventh row are 1, 6, 15, 20, 15, 6, 1, these are the coefficients of the binomial expansion of (a + b)⁶.

4. Using these coefficients the following binomials can be explained as

$(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}$

$(a+b)^{4}=a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}$

$(a+b)^{5}=a^{5}+5a^{4}b+10a^{3}b^{2}+10a^{2}b^{3}+5ab^{4}+b^{5}$

$(a+b)^{6}=a^{6}+6a^{5}b+15a^{4}b^{2}+20a^{3}b^{3}+15a^{2}b^{4}+6ab^{5}+b^{6}$

$(a+b)^{7}=a^{7}+7a^{6}b+21a^{5}b^{2}+35a^{4}b^{3}+35a^{3}b^{4}+21a^{2}b^{5}+7ab^{6}+b^{7}$

And so on we can expand the other binomials with different exponents.

Result

With the help of Pascal’s triangle we can easily expand the binomial with any positive integral exponent.

Applications

1. This activity can be used to write binomial exponent for (a + b)ⁿ, where n is a positive integer.

2. Pascal Triangle also explain the concept of combinations.

VIVA – VOICE

Q. 1. What is a monomial ?

Ans. An algebraic expression consisting of one element is called monomial.

Q. 2. What is a binomial?

Ans. An algebraic expression consisting of two elements is called binomial.

Q. 3. What would be the number of terms in the expression of (a+b)ⁿ.

Ans. 22 terms

Q. 4. What is the Pascal’s triangle?

Ans. In mathematics Pascal triangle is a triangular array of the binomial coefficients.

Q. 5. What is the importance of Pascal’s Triangle?

Ans. Pascal’s triangle is important because it is a never ending equilateral triangle of numbers that follow a rule of adding the two numbers above to get the number below.

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