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\((a+b)^n=\displaystyle\sum_{k=0}^n\binom nka^{n-k}b^k\)

where \(\dbinom nk=\dfrac{n!}{k!(n-k)!}\). The third term of the expansion occurs when \(k=2\).

So the third term of the expansion of \((3x^2+2y^3)^4\) is

\(\dbinom42(3x^2)^{4-2}(2y^3)^2=6(3^2x^4)(4y^6)=216x^4y^6\)

The coefficient is 216.

Answer;

-The coefficient of the third term in the expansion of the binomial

(3x^2 +2y^3)^4 is 216.

Solution;

Expand the binomial (3x² + 2y³)^4

Coefficients for expansion to power 4 are; 1, 4, 6, 4, 1

Thus; (3x² + 2y³)^4

=1(3x²)^0(2y³)^4 + 4 (3x²)^1(2y³)³ + 6 (3x²)² (2y³)² + 4 (3x²)³(2y³) + 1(3x²)^4(2y³)^0

=81 x^8 +216 x^6 y^3 +216 x^4 y^6 +96 x^2 y^9 +16 y^12

The third term is 216 x^4 y^6.

The coefficient is 216.


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