The value of $\sum\limits_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is

The common difference of the A.P.: $a_1, a_2, \ldots, a_{\mathrm{m}}$ is 13 more than the common difference of the A.P.: $b_1, b_2, \ldots, b_n$. If $b_{31}=-277, b_{43}=-385$ and $a_{78}=327$, then $a_1$ is equal to

The value of

$$ \lim\limits_{x \rightarrow 0} \frac{\log _e\left(\sec (e x) \cdot \sec \left(e^2 x\right) \cdot \ldots \cdot \sec \left(e^{10} x\right)\right)}{e^2-e^{2 \cos x}} $$

is equal to

Let $\mathrm{S}=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9 -digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,