Question Number 178021 by Acem last updated on 12/Oct/22
![Let S={1, 2, 3, 4, 5} , we want to make a group H of numbers have the following properties: 1. Each number has different digits and taken from S 2. Each number is greater than 20 000 3. None of them is multiple of five How many items of H?](https://www.tinkutara.com/question/Q178021.png)
Answered by mr W last updated on 12/Oct/22
![4×4×3×2×1−3×3×2×1=78](https://www.tinkutara.com/question/Q178028.png)
Commented by Tawa11 last updated on 12/Oct/22
![Great sir](https://www.tinkutara.com/question/Q178035.png)
Commented by aurpeyz last updated on 12/Oct/22
![thank you Sir. I got how you handled the first two but i didnt gwt how you did that of the multiples of 5. pls explain](https://www.tinkutara.com/question/Q178040.png)
Commented by mr W last updated on 12/Oct/22
![let′s begin with the last digit, which should not be 5. so we have 4 ways to select the last digit. for the second last digit we have also 4 possiblities and for the next digit 3 possibilities etc. so tatally we have 4×4×3×2×1 ways. among them some have 1 as the first digit. these numbers are not valid, because they are smaller than 20000. now we shoud find how mang numbers begin with 1. 1abcd abcd are from 2,3,4,5. similary as above we can get that there are 3×3×2×1 such numbers. so the answer is 4×4×3×2×1−3×3×2×1=78.](https://www.tinkutara.com/question/Q178042.png)
Commented by Acem last updated on 12/Oct/22
![I appreciate the professor′s method, it′s very clear and easy. I was going to mention mine but it′s a bit complicated. Big thanks](https://www.tinkutara.com/question/Q178053.png)
Commented by aurpeyz last updated on 12/Oct/22
![thank you so much](https://www.tinkutara.com/question/Q178057.png)
Commented by Acem last updated on 12/Oct/22
![A simple explanation of the almighty professor′s method a= 20 000 determinant ((,,,,(Not 5),(NW_(Num_s >a or Num_s <a) = 96)),(1,2,3,4,( 4),(← Start from here))) determinant ((( 1),,,,(Not 5),(NW_(Num_s < a) = 18)),(−,1,2,3,( 3),(← Start from here))) NumItems_H = the difference= 78 numbers](https://www.tinkutara.com/question/Q178069.png)
Commented by aurpeyz last updated on 12/Oct/22
![nice one](https://www.tinkutara.com/question/Q178070.png)