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 $B = 21 + 6 + 1 = 28$
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