Consider the matrix

$$ M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}. $$

Let $p, q, r, s, a, b, c$ and $d$ be integers such that

$$ M^{26} = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \quad \text{and} \quad \sum\limits_{k=1}^{26} M^k = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. $$

Then which of the following statements is (are) TRUE?

Let $S = \{1, 2, 3, \ldots, 10\}$. Consider the set

$X = \{R : R \text{ is an equivalence relation on the set } S \text{ such that } R \text{ has exactly 42 elements}\}$.

Then the number of elements in $X$ is ____________.

Consider the function $f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by

$$f(x) = (|x| + |x-1|) \sin x + \left[ x \sin x \right],$$

where $\left[ x \sin x \right]$ is the greatest integer less than or equal to $x \sin x$.

Let $\alpha$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT differentiable.

Then the value of $\alpha + \beta$ is ____________.

The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is ______________.