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If the coefficient of \( x \) in the expansion of \( (a x^2 + b x + c)(1 - 2x)^{26} \) is \( -56 \) and the coefficients of \( x^2 \) and \( x^3 \) are both zero, then \( a + b + c \) is equal to:

Let the smallest value of \( k \in \mathbb{N} \), for which the coefficient of \( x^3 \) in \( (1+x)^3 + (1+x)^4 + (1+x)^5 + \ldots + (1+x)^{99} + (1+kx)^{100} \), \( x \neq 0 \), is \( \left( \frac{43n + 101}{4} \right)\binom{100}{3} \) for some \( n \in \mathbb{N} \), be \( p \). Then the value of \( p + n \) is:

The sum of all possible values of \( n \in \mathbb{N} \), so that the coefficients of \( x \), \( x^{2} \) and \( x^{3} \) in the expansion of \( (1 + x^{2})(1 + x)^{n} \), are in arithmetic progression is:

If the sum of the coefficients of \( x^{7} \) and \( x^{14} \) in the expansion of \( \left(\frac{1}{x^{3}}-x^{4}\right)^{n} \), \( x\neq 0 \), is zero, then the value of \( n \) is ______.

If for then \( m \) equals:

If the coefficients of the middle terms in the binomial expansions of \( (1+\alpha x)^{26} \) and \( (1-\alpha x)^{28} \), \( \alpha \neq 0 \) are equal, then the value of \( \alpha \) is:

Let \( S = \dfrac{1}{25!} + \dfrac{1}{3!23!} + \dfrac{1}{5!21!} + \ldots \) upto \( 13 \) terms. If \( 13S = \dfrac{2^{k}}{n!} ,\ k \in \mathbb{N} \), then \( n + k \) is equal to:

The coefficient of \( x^2 \) in the expansion of \( \left(2x^2 + \frac{1}{x}\right)^{10} \), \( x \neq 0 \), is:

In the expansion of \( \left(9x-\frac{1}{3\sqrt{x}}\right)^{18},\ x>0 \), if the term independent of \( x \) is \( (221)k \), then \( k \) is equal to:

If \( (1-x^{3})^{10}=\sum_{r=0}^{10} a_{r}x^{r}(1-x)^{30-2r} \), then \( \dfrac{9a_{9}}{a_{10}} \) is equal to __________

The value of \( \frac{{}^{100}C_{50}}{51} + \frac{{}^{100}C_{51}}{52} + \ldots + \frac{{}^{100}C_{100}}{101} \) is:

If \( 26\left( \dfrac{2^{3}}{3}\left({}^{12}C_{2}\right)+\dfrac{2^{5}}{5}\left({}^{12}C_{4}\right)+\dfrac{2^{7}}{7}\left({}^{12}C_{6}\right)+\ldots+\dfrac{2^{13}}{13}\left({}^{12}C_{12}\right) \right)=3^{13}-a \), then \( a \) is equal to:

The coefficient of \( x^{48} \) in \( (1 + x) + 2(1 + x)^{2} + 3(1 + x)^{3} + \ldots + 100(1 + x)^{100} \) is: