Question 63

If $$A$$ denotes the sum of all the coefficients in the expansion of $$(1 - 3x + 10x^2)^n$$ and $$B$$ denotes the sum of all the coefficients in the expansion of $$(1 + x^2)^n$$, then :

To find the sum of all coefficients in a polynomial expansion, substitute $$x = 1$$.

$$A = (1 - 3(1) + 10(1)^2)^n = (1 - 3 + 10)^n = 8^n$$

$$B = (1 + (1)^2)^n = 2^n$$

Now, $$A = 8^n = (2^3)^n = 2^{3n} = (2^n)^3 = B^3$$.

The answer is $$A = B^3$$, which corresponds to Option (1).

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