<s>[INST] Finding the number of $4$-digit numbers so no two adjacent digits are even Find the number of $4$-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even.
 [/INST] Even digits $\{0,2,4\}$
Odd digits $\{1,3,4\}$
Numbers of the format: Even Odd Even Odd

$2 \times 3 \times 2 \times 2=24$

Numbers of the format: **Odd Even Odd Even

$3 \times 3 \times 2 \times 2=36$

Numbers of the format:  Odd Odd Odd Even

$3 \times 2 \times 1 \times 3=18$

Numbers of the format: Odd Odd Even Odd 

$18$

Numbers of the format: Odd Even Odd Odd 

$18$

Numbers of the format: Even Odd Odd Odd 

$2 \times 3 \times 2 \times 1=12$

**Numbers of the format: Even Odd Odd Even ** 

$2 \times 3 \times 2 \times 2=24$

Required count: 

$24+36+18+18+18+12+24=150$

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