## mathmath333 one year ago Counting Problem

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{In how many ways can 4 boys & 4 girls be arranged}\hspace{.33em}\\~\\ & \normalsize \text{such that no 2 girls are together.}\hspace{.33em}\\~\\ \end{align}}

2. anonymous

You can probably make an organized list, or sample space

3. mathmath333

ok

4. mathmath333

3 girls together also includes 2 girls together

5. mathmath333

6. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{In how many ways can 4 boys & 4 girls be arranged in a row}\hspace{.33em}\\~\\ & \normalsize \text{such that no 2 girls are together.}\hspace{.33em}\\~\\ \end{align}}

7. ganeshie8

forget about girls, first you arrange the $$4$$ boys

8. ganeshie8

how many ways can you arrange $$4$$ boys ?

9. mathmath333

4!=24

10. ganeshie8

Yes |dw:1440079075919:dw|

11. anonymous

don't mind me - I am not good with this stuff - just read/listen to what @ganeshie8 wrote

12. mathmath333

5*4=20

13. IrishBoy123

.

14. ganeshie8

scratch that

15. mathmath333

have u also counted this situation gbbgbgbg

16. ganeshie8

next, for each of that arrangement, the girls can only go between the boys. so there are exactly $$5$$ for the $$4$$ girls to go : |dw:1440079754229:dw|

17. ganeshie8

how many ways can you arrange 4 girls in 5 chairs ?

18. IrishBoy123

.

19. ganeshie8

In other words, how many 4 letter words can you make using 5 different letters ?

20. mathmath333

4!

21. ganeshie8

no, try again

22. mathmath333

5!

23. ganeshie8

yes so the final answer is $$4!*5!$$

24. mathmath333

is their any alternate way for this

25. ganeshie8

there are many ways to work it the one i showed you is the most simplest one

26. mathmath333

example using negation of sentense "no 2 girls are together"

27. ganeshie8

that would only complicate things because you need to consider different cases : 2 girls together, 3 girls together, 4 girls together

28. ganeshie8

the method i showed you is simple : 1) first place boys 2) next work out where girls can go