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
+20480v⁴
s+5120v³
s²
+640v²
s³
+40vs⁵
+s⁵
.
