MonotonicityHard
Question
The number of strictly increasing functions f from the set $\{ 1,2,3,4,5,6\}$ to the set $(1,2,3,\ldots,9)$ such that $f(i) \neq i$ for $1 \leq i \leq 6$, is equal to :
Options
A.21
B.27
C.22
D.28
Solution
$f(i) \neq i,f(x)$ is strictly increasing function $f:A \rightarrow B$, where $A\{ 1,2,3\ldots\ldots.6\}$
B $\{ 1,2,3,\ldots\ldots 9\}$, then number of functions $f:A \rightarrow B$ is equal to
$$\begin{matrix} f(i) \neq i\text{~}\text{Case}\text{~} - if(1) & \ = 2 \Rightarrow \ ^{7}C_{5} = 21 \\ \text{~}\text{Case- ii}\text{~}f(1) & \ = 3 \Rightarrow \ ^{6}C_{5} = 6 \\ \text{~}\text{Case- iii}\text{~}f(1) & \ = 4 \Rightarrow \ ^{5}C_{5} = 1 \end{matrix}$$
No of function A to $B = 21 + 6 + 1 = 28$
Create a free account to view solution
View Solution FreeMore Monotonicity Questions
Function f(x) = is increasing when...In the following, decreasing function is...Consider f(x) = l 1 - x l , 1 ≤ x ≤ 2 and g(x) = f(x) + b sin π/2 x, 1 ≤ x ≤ 2 then which o...The function f(x) = cos x - 2px is monotonically decreasing for...If the domain of f(x) = sin x is = {x : 0 ≤ x ≤ π}, then f(x) is...