Given the polynomial expression (1+x+x2+x3)100, we are interested in the coefficient of xr, denoted by ar. To find the sum of the products râ‹…ar, we perform the following steps: First, evaluate the polynomial at x=1 : (1+1+12+13)100=(4)100 This shows that the sum of all coefficients,
300
∑
r=0
ar, equals 4100. Next, differentiate (1+x+x2+x3)100 with respect to x : ‌
d
dx
((1+x+x2+x3)100)=100(1+x+x2+x3)99(1+2x+3x2) Substitute x=1 into the derivative:
300
∑
r=0
r⋅ar‌=100(4)99(1+2⋅1+3⋅12) ‌=100(499)⋅6 Simplify the expression to find: