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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Algebra and Combination.

## Algebra and combination – AIME 2000

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

• is 107
• is 667
• is 840
• cannot be determined from the given information

Algebra

Equations

Combination

## Check the Answer

But try the problem first…

Source

AIME, 2000, Question 3

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

here coefficient of $x^{2}$= coefficient of $x^{3}$ in the same expression

Second Hint

then ${2000 \choose 1998}a^{2}b^{1998}$=${2000 \choose 1997}a^{3}b^{1997}$

Final Step

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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