Answer :
To expand [tex]\((x+y)^7\)[/tex] using Pascal's Triangle, we'll find the coefficients for each term in the expansion using the binomial coefficients.
1. Understanding Pascal's Triangle:
Pascal's Triangle provides the coefficients for the expansion of a binomial expression [tex]\((x+y)^n\)[/tex]. For [tex]\((x+y)^7\)[/tex], we'll refer to the 7th row (starting with row 0 being the topmost row, which is 1).
2. 7th Row of Pascal's Triangle:
The coefficients in the 7th row are:
[tex]\(1, 7, 21, 35, 35, 21, 7, 1\)[/tex].
3. Using these coefficients to write the expansion:
- Each term in the expansion is in the form [tex]\({n \choose k} x^{n-k} y^{k}\)[/tex], where [tex]\({n \choose k}\)[/tex] is the binomial coefficient.
- For [tex]\((x+y)^7\)[/tex], [tex]\(n = 7\)[/tex].
Thus, the expansion is:
[tex]\[
x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7
\][/tex]
4. Insert the missing coefficients into the expression:
Given expression:
[tex]\[
x^7 + 7x^6y + \square x^5y^2 + 35x^4y^3 + 35x^3y^4 + \square x^2y^5 + 7xy^6 + y^7
\][/tex]
Filling in the missing coefficients 21:
[tex]\[
x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7
\][/tex]
So the complete expansion of [tex]\((x+y)^7\)[/tex] is:
[tex]\[
x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7
\][/tex]
1. Understanding Pascal's Triangle:
Pascal's Triangle provides the coefficients for the expansion of a binomial expression [tex]\((x+y)^n\)[/tex]. For [tex]\((x+y)^7\)[/tex], we'll refer to the 7th row (starting with row 0 being the topmost row, which is 1).
2. 7th Row of Pascal's Triangle:
The coefficients in the 7th row are:
[tex]\(1, 7, 21, 35, 35, 21, 7, 1\)[/tex].
3. Using these coefficients to write the expansion:
- Each term in the expansion is in the form [tex]\({n \choose k} x^{n-k} y^{k}\)[/tex], where [tex]\({n \choose k}\)[/tex] is the binomial coefficient.
- For [tex]\((x+y)^7\)[/tex], [tex]\(n = 7\)[/tex].
Thus, the expansion is:
[tex]\[
x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7
\][/tex]
4. Insert the missing coefficients into the expression:
Given expression:
[tex]\[
x^7 + 7x^6y + \square x^5y^2 + 35x^4y^3 + 35x^3y^4 + \square x^2y^5 + 7xy^6 + y^7
\][/tex]
Filling in the missing coefficients 21:
[tex]\[
x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7
\][/tex]
So the complete expansion of [tex]\((x+y)^7\)[/tex] is:
[tex]\[
x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7
\][/tex]
