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 $f(x_i)\ne y_i\forall i=(1,2,\mathrm{3....}r)$, 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
$=r!\left(1-\frac{1}{1}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+...+\frac{{\left(-1\right)}^r}{r!}\right).....(\ast )$
$\therefore$ Putting $r=4$ in $(\ast )$, we get
$t_4=4!\left(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\right)$
$=4!\left(\frac{1}{2}-\frac{1}{6}+\frac{1}{24}\right)$
$=\frac{4!}{4!}\left[12-4+1\right]=9$