Find the color(blue)5th term of (4x-y)^color(red)8
Examine Pascal's triangle below, and find the color(red)8th row where the first row is row "zero". Use the color(red)8th row because the exponent on the binomial is color(red)8.
The color(red)8th row is color(red)(1, 8, 28, 56, )color(blue)(70), color(red)(56, 28, 8, 1 ).
Because we are looking for the color(blue)5th term, use the color(blue)5th number in this row, which is color(blue)(70).
The number color(blue)(70) is used as a preliminary coefficient of the color(blue)5th term.
The next part of the color(blue)5th term is first term of the binomial (4x) raised to the exponent color(violet)4 as found by counting down from 8 from the left.
The last part of the color(blue)5th term is the 2nd term of the binomial (-y) raised to the exponent color(limegreen)4 as found by counting down from 8 from the right.
Note that the two exponents, color(violet)4 and color(limegreen)4, should add up to color(red)8 because the exponent on the original binomial is color(red)8.
The entire term is color(blue)(70) (4x)^color(violet)4 (-y)^color(limegreen)4=70(256x^4)(y^4)=17920x^4y^4
color(white)(AaaaAAaaaAAAAAA)1
color(white)(AAAaAaA^2AAaA)1color(white)(aa)1
color(white)(aaaaaaaaaaa)1color(white)(aa)2color(white)(aa)1
color(white)(aaaaaaaaa)1color(white)(aa)3color(white)(aaa)3color(white)(aa)1
color(white)(aaaaaaa)1color(white)(aa)4color(white)(aaa)6color(white)(aaa)4color(white)(aa)1
color(white)(aaaaa)1color(white)(aaa)5color(white)(aa)10color(white)(aa)10color(white)(aa)5color(white)(aa)1
color(white)(aaa)1color(white)(aaa)6color(white)(aa)15color(white)(aa)20color(white)(aa)15color(white)(aa)6color(white)(aa)1
color(white)(aa)1color(white)(aa)7color(white)(aa)21color(white)(aa)35color(white)(aa)35color(white)(aa)21color(white)(aa)7color(white)(aa)1
color(white)1color(red)1color(white)(a^2)color(red)8color(white)(a^2)color(red)(28)color(white)(aa)color(red)56color(white)(aa)color(blue)(70)color(white)(aa)color(red)(56)color(white)(aa)color(red)(28)color(white)(a^2)color(red)8color(white)(aa)color(red)1
uarrcolor(white)(a)uarrcolor(white)auarrcolor(white)auarrcolor(white)(aa)uarrcolor(white)(aa)uarrcolor(white)(aa)uarrcolor(white)(a)uarrcolor(white)auarr
color(white)(a)8color(white)(aa)7color(white)(aa)6color(white)(a^2a)5color(white)(aaa)color(violet)4color(white)(aaa)3color(white)(aaa)2color(white)(aaa)1color(white)(aa)0color(white)(a)larr1st exponent
color(white)(a)0color(white)(aa)1color(white)(aa)2color(white)(a^2a)3color(white)(aaa)color(limegreen)4color(white)(aaa)5color(white)(aaa)6color(white)(aaa)7color(white)(aa)8color(white)(a)larr2nd exponent
