## mathmath333 one year ago Counting Problem

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{How many words of 11 letters could be formed with } \hspace{.33em}\\~\\ & \normalsize \text{all the vowels present in even places, using all the } \hspace{.33em}\\~\\ & \normalsize \text{letters of the alphabet ?(without repetition) } \hspace{.33em}\\~\\ \end{align}}

2. hartnn

what are your thoughts?

3. mathmath333

\large \color{black}{\begin{align} & a.)\ ^{21}P_{6}\times 5! \hspace{.33em}\\~\\ & b.)\ 21! \hspace{.33em}\\~\\ & c.)\ ^{21}P_{5}\times 5! \hspace{.33em}\\~\\ & d.)\ ^{26}P_{8} \hspace{.33em}\\~\\ \end{align}}

4. mathmath333

|dw:1440438880520:dw|

5. mathmath333

i m confused on how to think here

6. hartnn

those 5 places are blocked with 5 vowels, which can be arranged among themselves in how many ways??

7. mathmath333

6 letters should be vowels and 5 letters non vowels

8. hartnn

how many other letters are left? in how many ways can you arrange those other letters in 6 places?

9. hartnn

5 vowels, at places: 2,4,6,8,10

10. mathmath333

\large \color{black}{\begin{align} & ^{6}P_{5}\ ways \hspace{.33em}\\~\\ \end{align}}

11. mathmath333

iis it correct

12. hartnn

sorry for asking many questions at one time. lets just concentrate on 5 places are blocked with 5 vowels, which can be arranged among themselves in how many ways??

13. mathmath333

5! ways

14. hartnn

good! now forget about those 5 places, they are done only 6 more places to go, and how many options do we have? hint : consonants

15. mathmath333

there are 6 places and 21 consonants remaining

16. hartnn

and how can we arrange that?

17. mathmath333

\large \color{black}{\begin{align} & ^{21}P_{6}\ ways \hspace{.33em}\\~\\ \end{align}}

18. hartnn

$$\huge \checkmark$$

19. hartnn

any more doubts? :)

20. hartnn

yes

21. mathmath333

does u mean 1st option is correct

22. mathmath333

ok thanks

23. hartnn

yes. clear any doubts that you have in your mind ...