Combinatorial Probability?

Question

If no digits can be repeated, how many 9-digit numbers can be made from the set {1 2 3 4 5 6 7 8 9} so that:
a) Every even digit has an odd digit before and after it?
b) All the even digits remain adjacent? ( I don't understand what is meant by this question.)

Answer 1

a)
There are four even digits (2, 4, 6, 8) and five odd digits (1, 3, 5, 7, 9), so the only way to arrange them is:

odd - even - odd - even - odd - even - odd - even - odd

Since order matters, we use permutations. The number of ways to arrange 4 objects is
4! = 4*3*2*1 = 24
The number of ways to arrange five objects is
5! = 5*4*3*2*1 = 120

We need to arrange 4 objects, and also 5 objects separately (none of them can go in the place of the other), so the number of permutations is simply
4! * 5! = 24 * 120 = 2880

b)
If all the even digits remain adjacent, the only ways to arrange them are these six ways:

EEEEOOOOO
OEEEEOOOO
OOEEEEOOO
OOOEEEEOO
OOOOEEEEO
OOOOOEEEE

where E is an even number and O is an odd number. Now, for each one of these, there are 2880 ways to arrange the numbers (from our previous calculation). So the total number of arrangements is

6 * 2880 = 17280

Answer 2

a) The first, third, fifth, 7th and last digits must be odd. There are 5 choices for the first of these, 4 left for the third, 3 for the fifth, 2 for the seventh and 1 left over for the last. So there are 5*4*3*2*1 = 120 ways to do this.

Similarly, the even digits (2, 4, 6, 8) must be in the other positions. So there are 4*3*2*1 = 24 ways to place them.

120*24 = 2880 ways overall.

b) The even digits must remain adjacent means that there cannot be ANY odd digits between the even digits. All four even digits must be together in the larger number.
There are still 24 ways to order the even digits.
Now consider this set of numbers together as if it were just one digit. Then the five odd digits plus this one unit of even digits makes 6 things that must be put in order. So there are 6*5*4*3*2*1 = 720 ways to do so.

720*24 = 17,280 ways to put them in order with the even numbers staying adjacent to each other.

I hope this helps!