## mathmath333 one year ago The number of ways in which four particular persons A,B,C,D and six more persons can stand in a queue so that A always stands before B, B before C, and C before D, is ?

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{ The number of ways in which four particular persons A,B,C,D }\hspace{.33em}\\~\\ & \normalsize \text{ and six more persons can stand in a queue so that A always }\hspace{.33em}\\~\\ & \normalsize \text{ stands before B, B before C, and C before D, is ? }\hspace{.33em}\\~\\ \end{align}}

2. dan815

|dw:1442643874411:dw|

3. mathmath333

this is one of the case

4. mathmath333

|dw:1442644105548:dw|

5. dan815

10!/4!

6. dan815

there are 10! ways to arrange these 10 persons now for each of those arrangements there is some arrangement of ABCD 4! ways of them, we only want the arrangement A B C D so 1 for every one of thsoe 4! ways thus 10!/4!

7. mathmath333

but the ABCD can be in many ways like this |dw:1442644275123:dw|

8. dan815

do you want clearer explaination

9. dan815

for example lets take some random arrangement in 10! 1 2 3 4 A 5 B 6 C D okay for this arrangement there are 4! other ways we count when A B C D can be placed in way , and the numbers 1 to 6 are left in same spot

10. mathmath333

is this (below) a valid set up|dw:1442644423498:dw|

11. dan815

we count that in the 10!

12. dan815

so taking 1 for every 4! we will take only the A B C D case for each of those arrangements in 10!

13. mathmath333

ok

14. dan815

15. mathmath333

16. dan815

ill make it clearer okay

17. dan815

for 10! ways we will have the arrangement of every single possible way of placing these 10 people

18. dan815

let us look at some of these arragenment sequences

19. dan815

|dw:1442644766778:dw|

20. dan815

there are 4! ways for this one arrangement of how you play these 6 people in the 10 spots

21. dan815

out of these 4! ways , we only want one of them where its A B C D

22. dan815

so another way less intuitive would now be this means all we care about is how we place the 6 people in the 10 spots

23. mathmath333

ok i got it

24. dan815

ok

25. mathmath333

|dw:1442645081498:dw|

26. dan815

welcome!