csat-suneung 2023 Q30

csat-suneung · South-Korea · csat__math 4 marks Permutations & Arrangements Counting Functions with Constraints

For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points] (가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$. (나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$. (다) $f ( 6 ) = f ( 5 ) + 6$

For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points]\\
(가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$.\\
(나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$.\\
(다) $f ( 6 ) = f ( 5 ) + 6$

Paper Questions

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