How many five digit numbers can be formed that are both odd and less than $40,000$ with the set $\{2,3,4,5,6\}$

I have to find how many possible five digit numbers which are both less than 40,000 and odd can be made with the set $$\{2,3,4,5,6\}$$ by using each number once only .

So far I have noted that there are only two possible choices for the first digit; $$2,3$$

And only two possible choices for the last digit; $$3,5$$

Hence, by using dashes to represent the possible number of choices for each corresponding digit:

$$\underline{2} \space \underline{}\space \underline{}\space \underline{}\space \underline{2}$$

Filling in the remaining three middle digits:

$$\underline{2} \space \underline{3}\space \underline{2}\space \underline{1}\space \underline{2}$$ $$2*3*2*1*2=24$$ However I know this is incorrect, I am a bit confused on how to deal with the fact that both the first digit and the last digit have an overlapping choice of the number $$3$$, I have tried to split it into two cases with one case where the first digit is $$2$$ and the other case where the first digit is $$3$$ but it

How would I go about breaking down this problem logically and solving it?

• @AlbusDumbledore My bad, I have corrected it. Dec 21, 2020 at 17:35
• Also i guess repeitions are allowed Dec 21, 2020 at 17:35
• @AlbusDumbledore They are not, I have added that additional info to the post Dec 21, 2020 at 17:41
• not "statistics" tag Dec 21, 2020 at 17:45

Assuming you cannot repeat digits...

Case 1: the first digit is a 2.

There are 2, candidates for the last digit, and 6 ways to fill the remaining digits.

12 arrangements under this condition.

Case 2: the first digit is a 3.

The last digit is a 5. There are still 6 ways to fill the remaining digits.

$$6+12 = 18$$

So you have find how many possible five digit numbers which are both less than 40,000 and odd can be made with the set {2,3,4,5,6}.Also considering that no numbers can be repeated.

So the possibilities for the first blank is 2 and 3, as they must be less than 40,000.

The second blank:4 The third blank:3 The fourth blank:2 The fifth blank has to be odd so it is 2.

So, you will get 243*2 =48 .