00:01
In this problem, our job is to find the number of four -digit numbers that satisfy the given conditions.
00:07
So we will write the four -digit number in the form abcd, and we will translate each of the conditions.
00:14
We should note that none of the digits are zero, and the digits are distinct, so we can't have any repeats there.
00:26
Alright, well we are told that the sum of the ones and tens of digits equals the sum of the other two.
00:36
2.
00:39
We are told that the sum of the tens and hundreds of digits is 3 times the sum of the other 2.
00:49
We are also told that if we add up all 4 digits, we cannot go over 20.
00:57
So let's see how we can combine these conditions and narrow down the number of possibilities.
01:07
So we're going to take each of those equations and solve for b.
01:14
So in that first equation, equation, b will be c plus d minus a.
01:19
On the second equation, we will have b equals 3a plus 3d minus c.
01:32
So right off the bat, we can set these expressions equal to each other.
01:39
So we have c plus d minus a equals 3a plus 3d minus c.
01:47
So we can solve for c.
01:49
We have have 2c is 4a plus 2d.
01:55
So c is 2a plus d.
01:59
All right, well, let's pause for a second and organize what we've got.
02:03
So far, we've expressed c in terms of a and d.
02:07
Now let's go back and see if we can express b in terms of a and d by using what we just got for c.
02:17
It's a little bit more algebra right here.
02:19
If b is c plus d minus a, then b must be 2a plus d plus d minus a.
02:35
So b turns out to be a plus 2d.
02:40
Some nice symmetry there between b and c.
02:44
All right.
02:47
Well, that helps us because we also so note that the sum of all four of these can't be bigger than 20.
02:53
So let's substitute what we have and see if that helps us here.
02:56
We know that a plus b, here's b now, plus c plus d cannot be bigger than 20.
03:10
Well, we now have four a's and four d's.
03:17
That sum has to be less than or equal to 20.
03:19
So now we know that a plus d is less than or equal to 5.
03:23
That is a big help.
03:25
That means that neither a nor d can get too big, and that's going to narrow down our number of possibilities here.
03:32
All right, so here we go.
03:34
We're going to use that condition.
03:37
We're going to use this expression and this expression to narrow down how many possible four -digit numbers satisfy this.
03:48
So if a plus d is less than or equal to 5, a and d can't be 0, and they have to be distinct, distinct, then one possibility is that a is 1 and d is 2.
03:57
In that case, you can go back and check and see that b, remember b is a plus 2d, so b will be 5, c is 2a plus d, so c will be 4, and so one of our numbers is 1, 5, 4, 2, writing that in the order a, b, you can check that real quick.
04:26
The sum of those digits does not exceed 20.
04:28
4 plus 2 equals 1 plus 5, and 5 plus 4 is 3 times 1 plus 2...