Question

g The sixth row of Pascal's Triangle is: 1 6 15 20 15 6 1 (a) What is the 7th row of Pascal's Triangle? (b) Use your answer to the previous problem to write the expanded form of (x + y)7.

Answer 1

ANSWER

a)The 7th row is:

1 7 21 35 35 35 21 7 1

b)

${(x + y)}^{7} = {x}^{7}+ 7 {x}^{6} y+ 21 {x}^{5} {y}^{2}+ 35 {x}^{4} {y}^{3} + 35{x}^{3} {y}^{4} + 21{x}^{2} {y}^{5} + 7x {y}^{6}+ {y}^{7}$

EXPLANATION

The sixth row of Pascal's Triangle is:

1 6 15 20 15 6 1.

We generate the 7th row by repeating the eXtreme 1s and adding the entries directly above to generate the entries within as show in the attachment.

The 7th row is:

1 7 21 35 35 35 21 7 1

b) We can use this to expand

${(x + y)}^{7}$

We know that the degree of x is going to decrease from left to right and the degree of y is going to increase from left to right.

${(x + y)}^{7} = 1 ({x}^{7} ) + 7( {x}^{6} y) + 21( {x}^{5} {y}^{2} ) + 35( {x}^{4} {y}^{3} ) + 35( {x}^{3} {y}^{4} ) + 21( {x}^{2} {y}^{5} ) + 7( x {y}^{6} ) + 1( {y}^{7} )$

This simplifies to,

${(x + y)}^{7} = {x}^{7}+ 7 {x}^{6} y+ 21 {x}^{5} {y}^{2}+ 35 {x}^{4} {y}^{3} + 35{x}^{3} {y}^{4} + 21{x}^{2} {y}^{5} + 7x {y}^{6}+ {y}^{7}$