Page 54 - Mathematics Class - XI

3. Write the numbers as follows:
1 (First Row)

1 1 (Second Row)
1 2 1 (Third Row)

1 3 3 1 (Fourth Row)
1 4 6 4 1 (Fifth Row)
1 5 10 10 5 1 (Sixth Row)
1 6 15 20 15 6 1 (Seventh Row)

1 7 21 35 35 21 7 1 (Eighth Row)
4. Clearly to write binomial expansion of (x + y) , use the numbers given in the (n + 1) row.
th
n
DEMONSTRATION
1. Fig. (a) looks like a triangle and is referred to as Pascal’s Triangle.
2. Number in the first row gives the binomial expansion of (x + y) .
0
Numbers in the second row give the coefficients of the terms of the binomial expansion of (x + y) .
1
Numbers in the third row give the coefficients of the terms of the binomial expansion of (x + y) .
2
Numbers in the fourth row give the coefficients of the terms of the binomial expansion of (x + y) and
3
so on.

OBSERVATION

1. Numbers in the fifth row are 1, 4, 6, 4, 1, which are coefficients of the binomial expansion of (x + y) .
4
2. Numbers in the seventh row are 1, 6, 15, 20, 15, 6, 1, which are coefficients of the binomial expansion of
(x + y) .
6
3. (x + y) = x + 3x y + 3xy + y 3
2
3
2
3
4. (x + y) = x + 5x y + 10x y + 10x y + 5xy + y 5
4
4
2 3
3 2
5
5
5. (x + y) = x + 6x y + 15x y + 20x y + 15x y + 6xy + y 6
6
4 2
2 4
5
3 3
6
5
6. (x + y) = x + 8x y + 28x y + 56x y + 70x y + 56x y + 28x y + 8xy + y 8
4 4
3 5
8
7
2 6
7
8
6 2
5 3
CONCLUSION
n
From this activity, we can write the binomial expansion for (x + y) = ∑ n C r x nr− y (where n is a positive integer).
r
n
r=0
APPLICATION
This activity can be used to write binomial expansion for (x + y) , where n is a positive integer.
n
Knowledge Booster
Pascal’s Triangle is a triangle of numbers where each 1
number is the two numbers directly above it added 1 1
together (except for the edges, which are all ‘1’). 1 2 1
Here, we have highlighted that 1 + 2 = 3 1 3 3 1
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