The number of bijective functions $f : \{1, 3, 5, 7, \ldots, 99\} \rightarrow \{2, 4, 6, 8, \ldots, 100\}$ such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \geq f(99)$ is:

Let $A$ and $B$ be non-empty sets in $\mathbb{R}$ and $f : A \to B$ be a bijective function.
Statement $1$ : $f$ is an onto function.
Statement $2$ : There exists a function $g : B \to A$ such that $f \circ g = I_B$.

Let $f: R \rightarrow R$ be defined as $f(x)=x^{4}$. Choose the correct answer.

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$,then the value of $\frac{1}{5!}(\beta-\alpha)$ is. . . . . .

Let $f: N \rightarrow N$ be defined by $f(n) = \begin{cases} \frac{n+1}{2}; & \text{if } n \text{ is odd} \\ \frac{n}{2}; & \text{if } n \text{ is even} \end{cases}$. Then $f$ is:

If $f: N \times N \rightarrow N$ defined by $f(m, n) = mn$,then $f$ is . . . . . . .