If $c_0, c_1, c_2, \ldots, c_n$ denote the coefficients in the expansion of $(1+x)^n$, then the value of $c_1 + 2c_2 + 3c_3 + \ldots + nc_n$ is

If $P_{n}$ denotes the product of the binomial coefficients in the expansion of $(1+x)^{n}$,then $\frac{P_{n+1}}{P_n}=$

If $n$ is a positive integer such that $n \ge 3$,then the value of the sum to $n$ terms of the series $1 \cdot n - \frac{(n - 1)}{1!} (n - 1) + \frac{(n - 1)(n - 2)}{2!} (n - 2) - \frac{(n - 1)(n - 2)(n - 3)}{3!} (n - 3) + \dots$ is:

If $C_r = ^{100}C_r$,then the value of $1 \cdot C_0^2 - 2 \cdot C_1^2 + 3 \cdot C_2^2 - 4 \cdot C_3^2 + \dots + 101 \cdot C_{100}^2$ is equal to:

Evaluate the sum: $\left( \binom{21}{1} - \binom{10}{1} \right) + \left( \binom{21}{2} - \binom{10}{2} \right) + \left( \binom{21}{3} - \binom{10}{3} \right) + \dots + \left( \binom{21}{10} - \binom{10}{10} \right) = $

The value of the sum $\left({ }^{n} C_{1}\right)^{2}+\left({ }^{n} C_{2}\right)^{2}+\left({ }^{n} C_{3}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}$ is