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)?

How many terms are there in the expansion of $${(1 + 2x + {x^2})^5} + {(1 + 4y + 4{y^2})^5}$$ ?