## mathmath333 one year ago Counting question

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{Find the number of ways in which the letters of the word MACHINE }\hspace{.33em}\\~\\ & \normalsize \text{can be arranged so that the vowels may occupy only odd positions. }\hspace{.33em}\\~\\ & a.)\ 4!\times 4! \hspace{.33em}\\~\\ & b.)\ ^{7}P_{3}\times 4! \hspace{.33em}\\~\\ & c.)\ ^{7}P_{4}\times 3! \hspace{.33em}\\~\\ & d.)\ \text{none of these} \hspace{.33em}\\~\\ \end{align}}

2. ganeshie8

|dw:1440622392209:dw|

3. ganeshie8

How many positions are there for the vowels to go to ?

4. mathmath333

4 position for vowels

5. ganeshie8

how many vowels are there ?

6. mathmath333

|dw:1440622528962:dw|

7. mathmath333

3 vowels

8. ganeshie8

4 positions 3 vowels first vowel can go into any one of the 4 positions : 4 ways after that, second vowel can go into any one of the remaining 3 positions : 3 ways after that, the third vowel can go into any one of the remaining 2 positions : 2 ways so total ways of arranging 3 vowels into 4 positions is 4*3*2

9. ganeshie8

we're done with vowels, next look at rest of the word

10. mathmath333

can u apply formula for that 4 positions 3 vowels

11. ganeshie8

sure, what formula ?

12. mathmath333

Permutation

13. ganeshie8

yes it is the permutation formula : $$4P3$$ i just elaborated for you

14. mathmath333

ohk

15. ganeshie8

16. mathmath333

3 positions 4 cnsonants

17. ganeshie8

really ? we have just used 3 positions for vowels right

18. ganeshie8

after placing 3 vowels, we see 4 empty positions, yes ?

19. mathmath333

|dw:1440623136170:dw|

20. ganeshie8

there are no restrictions on consonants they can go into odd positions that are not occupied by vowels too

21. ganeshie8

there are 3 even positions but there are 4 "empty" positions

22. mathmath333

sry 4 positions

23. ganeshie8

Yes 4 positions 4 consonants how many permutations ?

24. mathmath333

4!

25. ganeshie8

Yes

26. mathmath333

is option a.) correct

27. ganeshie8

you can arrange vowels in 4*3*2 ways and for each of that arrangement, you can arrange consonants in 4! ways so total arrangements is ?

28. mathmath333

4!*4!

29. ganeshie8

looks good!

30. mathmath333

thnks

31. perl

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