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)larr#1st 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)larr#2nd exponent
