00:01
We are asked how many numbers greater than 3000 but less than 10 ,000 can be made using the digits 1, 2, 3, 5, 7, and 9 with no repetition of those digits.
00:10
And for part 1, it's just those constraints.
00:17
So it has to be greater than 3000.
00:19
So that means it has to be at least a 4 -digit number.
00:23
And it has to be less than 10 ,000, which means that it can't be a 5 -digit number.
00:29
Number, because if it's five digits, the smallest digit that it can begin with is one.
00:35
The other digits are not zero, so therefore any five digit number or more would be greater than 10 ,000.
00:41
So this tells us that a number can only be four digits.
00:46
Since it has to be greater than 3 ,000, the first digit must be one of these four values.
00:52
If it's one of those four values, it's going to be more than 3 ,000, otherwise it would be less than 3 ,000.
00:57
So for the first digit, we have four possibilities, and then for the remaining three digits, it can be any three of the remaining five numbers.
01:17
So there are five, use three ways to pick three of the remaining five numbers, and then also the arrangement of these numbers matters because the order of the digits in a number matters.
01:29
So we multiply by three factorial.
01:33
So this is the number of ways of picking three of the remaining five digits.
01:36
And then for each of those combinations of three digits, there are three factorial ways to arrange them.
01:44
And so this product comes out to 240.
01:49
And then for part two, we're asked how many of those numbers would be even...