Find the number of all one-one functions from set $A = \{1, 2, 3\}$ to itself.

Let $R = \{ a, b, c, d, e \}$ and $S = \{1, 2, 3, 4\}$. The total number of onto functions $f: R \rightarrow S$ such that $f(a) \neq 1$ is equal to $.............$.

The number of onto functions from the set $\{1, 2, \ldots, 11\}$ to the set $\{1, 2, \ldots, 10\}$ is

If a set $A$ has $m$ elements and the set $B$ has $n$ elements,then the number of injections from $A$ to $B$ is

Show that the modulus function $f : R \rightarrow R$ given by $f(x) = |x|$ is neither one-one nor onto,where $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.

The number of strictly increasing functions $f$ from the set $A = \{1, 2, 3, 4, 5, 6\}$ to the set $B = \{1, 2, 3, \dots, 9\}$ such that $f(i) \neq i$ for all $1 \le i \le 6$ is equal to: