Question

Use Pascal’s Triangle to expand the binomial. (8v+s)^5

Answer 1

Pascal 's triangle is 

1
1,1
1,2,1
1,3,3,1
1,4,6,4,1
1,5,10,10,5,1

the row 1,5,10,10,5,1 is the one we need to expand (8v+s)⁵.

Given (a+b)⁵ each term of the row is the coefficient of akbt

with k goes form 5 to 0 and t goes from 0 to 5.

so (a+b)⁵=1a⁵b⁰+5a⁴b¹+10a³b²+10a²b³+5a¹b⁵+1a⁰b⁵

=a⁵+5a⁴b+10a³b²+10a²b³+5ab⁵+b⁵.

In the case of (8v+s)⁵:

a=8v
b=s

1(8v)⁵s⁰+5(8v)⁴s¹+10(8v)³s²+10(8v)²s³+5(8v)¹s⁵+1(8v)⁰s⁵

= (8v)⁵+5(8v)⁴s+10(8v)³s²+10(8v)²s³+5(8v)s⁵+s⁵=

= 8⁵v⁵+5⋅8⁴v⁴s+10⋅8³v³s²+10⋅8²v²s³+40vs⁵+s⁵=

= 32768v⁵+20480v⁴s+5120v³s²+640v²s³+40vs⁵+s⁵.