Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Algebra and Combination.
In expansion \((ax+b)^{2000}\) where a and b are relatively prime positive integers the coefficient of \(x^{2}\) and \(x^{3}\) are equal, find a+b
Algebra
Equations
Combination
Answer: is 667.
AIME, 2000, Question 3
Elementary Algebra by Hall and Knight
here coefficient of \(x^{2}\)= coefficient of \(x^{3}\) in the same expression
then \({2000 \choose 1998}a^{2}b^{1998}\)=\({2000 \choose 1997}a^{3}b^{1997}\)
then \(b=\frac{1998}{3}\)a=666a where a and b are relatively prime that is a=1,b=666 then a+b=666+1=667.
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