Question

If $t_4$ denotes the number of one to one functions from $(x_1,x_2,,...,x_r)$ to $(y_1,y_2,...,y_r)$ such that $f\left(x\right)\ne y_i,i=(1,2,3,...r)$ then $t_4$ equals

• A. $9!$
• B. ${}^{11}{C}_{11}$
• C. $9$
• D. $12$

Answer 1

Answer: (C)

$9$

1. concept :
2. { use formula given below to find the number of derangements of all n things arranged in a row.
   

number of derangements = $n!(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}..........(-1{)}^n\frac{1}{n!})$

}

now in this question $n=4$

$t_4=4!(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!})$

$t_4=4!(1-\frac{1}{1}+\frac{1}{2}-\frac{1}{6}+\frac{1}{24})$

$t_4=24\frac{(12-4+1)}{24}$

$t_4=9$