1. ## math league

Of the integers between 10^3 and 10^4 that have no repeated digit, how many have digits that increase from left to right?

2. ## Re: math league

Originally Posted by victorwen28
Of the integers between 10^3 and 10^4 that have no repeated digit, how many have digits that increase from left to right?
Any subset of four nonzero digits fits that description.
So $\binom{9}{4}=~?$

3. ## Re: math league

126

{1234, 1235, 1236, 1237, 1238, 1239, 1245, 1246, 1247, 1248, 1249, 1256, 1257, 1258, 1259, 1267, 1268, 1269, 1278, 1279, 1289, 1345, 1346, 1347, 1348, 1349, 1356, 1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1456, 1457, 1458, 1459, 1467, 1468, 1469, 1478, 1479, 1489, 1567, 1568, 1569, 1578, 1579, 1589, 1678, 1679, 1689, 1789, 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, 2389, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479, 2489, 2567, 2568, 2569, 2578, 2579, 2589, 2678, 2679, 2689, 2789, 3456, 3457, 3458, 3459, 3467, 3468, 3469, 3478, 3479, 3489, 3567, 3568, 3569, 3578, 3579, 3589, 3678, 3679, 3689, 3789, 4567, 4568, 4569, 4578, 4579, 4589, 4678, 4679, 4689, 4789, 5678, 5679, 5689, 5789, 6789}

4. ## Re: math league

But there is a condition that the digits that have to increase from left to right.

5. ## Re: math league

Is there a way to do it without listing all the possibilities?

6. ## Re: math league

Originally Posted by MaxJasper
126
{1234, 1235, 1236, 1237, 1238, 1239, 1245, 1246, 1247, 1248, 1249, 1256, 1257, 1258, 1259, 1267, 1268, 1269, 1278, 1279, 1289, 1345, 1346, 1347, 1348, 1349, 1356, 1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1456, 1457, 1458, 1459, 1467, 1468, 1469, 1478, 1479, 1489, 1567, 1568, 1569, 1578, 1579, 1589, 1678, 1679, 1689, 1789, 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, 2389, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479, 2489, 2567, 2568, 2569, 2578, 2579, 2589, 2678, 2679, 2689, 2789, 3456, 3457, 3458, 3459, 3467, 3468, 3469, 3478, 3479, 3489, 3567, 3568, 3569, 3578, 3579, 3589, 3678, 3679, 3689, 3789, 4567, 4568, 4569, 4578, 4579, 4589, 4678, 4679, 4689, 4789, 5678, 5679, 5689, 5789, 6789}

Originally Posted by victorwen28
But there is a condition that the digits that have to increase from left to right.
Example: Consider the set $\{2,4,1,6\}$, how many ways can you arrange that set into a four-digit number of increasing digits, (in the literature that is known as a strictly sorted integer).
So there is an one-to-one correspondent between the strictly sorted four digit integers and the four elements subsets of $\{1,2,3,4,5,6,7,8,9\}$

7. ## Re: math league

So do you have to count in the end?

8. ## Re: math league

Originally Posted by victorwen28
So do you have to count in the end?
If you simply want the answer the please GO AWAY. You are a parasite.

If you have read questions, post them.

But if you are simply after an answer, log off.

9. ## Re: math league

I want to understand how to do it, but I am not clear of your approach.

10. ## Re: math league

Originally Posted by victorwen28
I want to understand how to do it, but I am not clear of your approach.
O.K.
If I give your the set $\{7,3,6,1\}$ how many way can you arrange those four into a strictly sorted integer?
And what is it?

11. ## Re: math league

(1,3,6,7)
only one way.

12. ## Re: math league

Originally Posted by victorwen28
(1,3,6,7)
only one way.
Correct!