mathmath333
  • mathmath333
Counting question
Mathematics
katieb
  • katieb
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{Find the number of ways in which the letters of the word MACHINE }\hspace{.33em}\\~\\ & \normalsize \text{can be arranged so that the vowels may occupy only odd positions. }\hspace{.33em}\\~\\ & a.)\ 4!\times 4! \hspace{.33em}\\~\\ & b.)\ ^{7}P_{3}\times 4! \hspace{.33em}\\~\\ & c.)\ ^{7}P_{4}\times 3! \hspace{.33em}\\~\\ & d.)\ \text{none of these} \hspace{.33em}\\~\\ \end{align}}\)
ganeshie8
  • ganeshie8
|dw:1440622392209:dw|
ganeshie8
  • ganeshie8
How many positions are there for the vowels to go to ?

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mathmath333
  • mathmath333
4 position for vowels
ganeshie8
  • ganeshie8
how many vowels are there ?
mathmath333
  • mathmath333
|dw:1440622528962:dw|
mathmath333
  • mathmath333
3 vowels
ganeshie8
  • 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
ganeshie8
  • ganeshie8
we're done with vowels, next look at rest of the word
mathmath333
  • mathmath333
can u apply formula for that 4 positions 3 vowels
ganeshie8
  • ganeshie8
sure, what formula ?
mathmath333
  • mathmath333
Permutation
ganeshie8
  • ganeshie8
yes it is the permutation formula : \(4P3\) i just elaborated for you
mathmath333
  • mathmath333
ohk
ganeshie8
  • ganeshie8
what about the consonants
mathmath333
  • mathmath333
3 positions 4 cnsonants
ganeshie8
  • ganeshie8
really ? we have just used 3 positions for vowels right
ganeshie8
  • ganeshie8
after placing 3 vowels, we see 4 empty positions, yes ?
mathmath333
  • mathmath333
|dw:1440623136170:dw|
ganeshie8
  • ganeshie8
there are no restrictions on consonants they can go into odd positions that are not occupied by vowels too
ganeshie8
  • ganeshie8
there are 3 even positions but there are 4 "empty" positions
mathmath333
  • mathmath333
sry 4 positions
ganeshie8
  • ganeshie8
Yes 4 positions 4 consonants how many permutations ?
mathmath333
  • mathmath333
4!
ganeshie8
  • ganeshie8
Yes
mathmath333
  • mathmath333
is option a.) correct
ganeshie8
  • 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 ?
mathmath333
  • mathmath333
4!*4!
ganeshie8
  • ganeshie8
looks good!
mathmath333
  • mathmath333
thnks
perl
  • perl
*

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