The number of strictly increasing functions f from the set { 1 , 2 , 3 , 4 , 5 , 6 } to the set { 1 , 2 , 3 , … , 9 } such that f ( i ) i for 1 i 6 , is equal to [2026]

Question

The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq i$ for $1 \leq i \leq 6,$ is equal to [2026]

Smart Achievers · Accepted Answer

(1)

$f (i) \neq i, f (x) is strictly increasing function$

$f : A \to B,$ $where A = {1, 2, 3, \dots, 6}$

$B = {1, 2, 3, \dots, 9},$ then number of functions $f : A \to B$ is equal to

$f (i) \neq i$ $Case (i) f (1) = 2 \Rightarrow {}^{7}{C_{5}} = 21$

$Case (ii) f (1) = 3 \Rightarrow {}^{6}{C_{5}} = 6$

$Case (iii) f (1) = 4 \Rightarrow {}^{5}{C_{5}} = 1$

Number of functions from A to B $= 21 + 6 + 1 = 28$

Solution

Q.
The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq i$ for $1 \leq i \leq 6,$ is equal to [2026]

1 28

2 22

3 27

4 21

Want Us to Email you About Special Offers & Updates?