The coefficient of x⁹⁹ in the expansion of (x $$-$$ 1) (x $$-$$ 2) (x $$-$$ 3) ...... (x $$-$$ 100) is

What is $${}^{47}{C_4} + {}^{51}{C_3} + \sum\limits_{j = 2}^5 {{}^{52 - j}{C_3}} $$ equal to?

The number of terms in the expansion of (x + a)¹⁰⁰ + (x $$-$$ a)¹⁰⁰ after simplification is

In the expansion of (1 + x)⁵⁰, the sum of the coefficients of odd powers of x is