Let the coefficient of the middle term of the binomial expansion of (1 + x)²ⁿ be $$\alpha$$ and those of two middle terms of the binomial expansion of (1 + x)^{2n $$-$$ 1} be $$\beta$$ and $$\gamma$$. Which one of the following relations is correct?

In the expansion of (1 + x)⁴³, if the coefficients of (2r + 1)th and (r + 2)th terms are equal, then what is the values of r(r $$\ne$$ 1) ?

If the coefficients of a^m and aⁿ in the expansion of (1 + a)^{m + n} are $$\alpha$$ and $$\beta$$, then which one of the following is correct?

What is the number of non-zero terms in the expansion of $${(1 + 2\sqrt 3 x)^{11}} + {(1 - 2\sqrt 3 x)^{11}}$$ (after simplification)?