<span>Pascal 's triangle is
</span>
1<span>
</span><span><span>1,1</span><span>
</span></span><span><span>1,2,1</span><span>
</span></span><span><span>1,3,3,1</span><span>
</span></span><span><span>1,4,6,4,1</span><span>
</span></span><span>1,5,10,10,5,1</span>
the row <span>1,5,10,10,5,1</span> is
the one we need to expand <span><span><span>(8</span>v+s)</span></span>⁵.
Given <span><span>(a+b)</span></span>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 <span><span><span><span>(a+b)</span></span></span></span>⁵=1a⁵b⁰+5a⁴b¹+10a³b²+10a²b³+5a¹b⁵+1a⁰b⁵
<span><span>=<span>a</span></span></span>⁵+5a⁴b+10a³b²+10a²b³+5ab⁵+b⁵.
In the case of <span><span><span>(8</span>v+s)</span></span>⁵:
<span><span>a<span>=8</span>v</span><span>
</span></span><span>b=s</span>
<span><span>1<span><span>(8</span>v)</span></span></span>⁵s⁰+5(8v)⁴s¹+10(8v)³s²+10(8v)²s³+5(8v)¹s⁵+1(8v)⁰s⁵
<span>
<span><span><span>=<span><span> (8</span>v)</span></span></span></span>⁵+5(8v)⁴s+10(8v)³s²+10(8v)²s³+5(8v)s⁵+s⁵=</span>
<span><span><span>= 8</span></span></span>⁵v⁵+5⋅8⁴v⁴s+10⋅8³v³s²+10⋅8²v²s³+40vs⁵+s⁵=
<span><span><span>= 32768</span><span><span>v</span></span></span></span>⁵<span><span><span>+20480</span><span><span>v</span></span></span></span>⁴<span><span><span>s</span><span>+5120</span><span><span>v</span></span></span></span>³<span><span><span><span>s</span></span></span></span>²<span><span><span>+640</span><span><span>v</span></span></span></span>²<span><span><span><span>s</span></span></span></span>³<span><span><span>+40</span><span>v<span>s</span></span></span></span>⁵<span><span><span>+</span><span>s</span></span></span>⁵<span>.</span>
