Question

Find the number of ways of arranging the numbers 1,2,3,4,5,6,7, if no two even numbers can be adjacent, and no two odd numbers can be adjacent.

Answer 1

24 ways

Explanation:

1) In Combinatorics when we arrange a number and the order matter, we call it arrange the possibilities. For this exercise let's not use formulas but reasoning.

2) For this case we need a two figure number. Since we have seven numbers.

Since there is no repetition, all the possibilities are:

`7*6=42`

3) But there is a restriction it's forbidden adjacent even and odd numbers: These numbers we don't want them:

13 15 17

24 26

31 35 37

42 46

51 53 57

62 66

71 73 75

18 non desirable results

The total arrangements minus the not possible combinations, will match the possible results:

`42-18=24`

3) Just for checking, we have here the allowed combinations:

12 14 16

21 23 25 27

32 34 36

41 43 45 47

52 54 56

61 63 65 67

72 74 76

A total of 24 possible ways.