Question

If $t_r$ denotes the number of $1 - 1$ function from $\{x_1, x_2 ... x_r\}$ to $\{y_1, y_2, ... y_r\}$ such that $f(x_i) \ne y_i \,\forall i = (1,2,3....r )$, then $t_4$ equals

• A. $9!$
• B. $^{11}C_2 - 11$
• C. $9$
• D. None of these

Answer 1

Answer: (C)

The problem is just similar as, if we have $x$ letters and n envelops and no letter goes to the right envelops.
$t_r = $ number of ways of putting $x_i = 1,2,... r$, in $r$ places
so that no $x_i$ goes to the corresponding place
$= r!\left( 1 - \frac{1}{1} + \frac{1}{2!} -\frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + ... + \frac{\left(-1\right)^{r}}{r!}\right) .....(*)$
$\therefore $ Putting $ r = 4$ in $(*)$, we get
$t_{4} = 4!\left(\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}\right) $
$ = 4!\left(\frac{1}{2} - \frac{1}{6}+\frac{1}{24}\right)$
$ = \frac{4!}{4!}\left[12 - 4 + 1\right] = 9$