Would appreciate it if someone helps with 31 and 5c pls. Answers are: 2^n-2 and 108
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Sorry, forgot to get back to this.
The question's quite bizarre.
There has got to be an easier way of doing it. What I did was I tried to invent a method for part a), and then copy it but with the even number restriction for part c.
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For part a), we first note that every 5 digit number is greater than 4000. For {0,2,4,5,7}, the number of 5 digit numbers we can make (without repetition) is 4*4! = 96 because the first digit cannot be 0.
This leaves us with 72 possible 4 digit numbers greater than 4000. To check this, we may split the cases based off the issue that the first digit cannot be a 0 or 2.
Note: 0 and 2 cannot BOTH be not picked, because then our numbers are only 3 digits
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Nice. Now let's add in the restriction.
For 5 digit numbers that are even, the last digit must be 0, 2 or 4. Split the cases up.
108 - 60 = 48, so apparently there are 48 possible 4 digit numbers. Let's check this.
Problem: We have TWO restrictions on 0 and 2. They can't be the first digit, and they may be forced into being the last digit. This is going to look bizarre.
Instead, consider the complementary event, where the numbers are still greater than 4000 BUT they are ODD. Remember that from above, we know that the total number of 4 digit numbers is 72, so we expect that the total number of odd 4 digit numbers is 24.
Note that we actually don't need to use the complement, and the maths is POTENTIALLY easier (I have not tried it). But it will make the visualisation even harder.
Things go haywire, as we need to decide between {4,5,7} which get chosen.
Subcase 3.1: {4,5} chosen
5 has to be the last digit
4 has to be the first digit
Then 0 and 2 can be placed in any order - 2 possible ways
Subcase 3.2: {4,7} chosen
Just like above - 2 possible ways
Subcase 3.3: {5,7} chosen
5 or 7 is the last digit - 2 possible ways
The leftover has to be the first digit
0 and 2 can still be placed in any order - 2 possible ways
So this subcase yields 2*2 = 4 possible ways
Try repuzzling everything from here.