# In how many different ways can the letters of the word MACHINE be arranged so that the vowels may occupy only the odd positions?

A. 210

B. 576

C. 144

D. 1728

E. 3456

### Solution(By Examveda Team)

There are 7 letters in the given word, out of which there are 3 vowels and 4 consonants.
Let us mark the positions to be filled up as follows:
$$\left( {\mathop {}\limits^1 } \right)\left( {\mathop {}\limits^2 } \right)\left( {\mathop {}\limits^3 } \right)\left( {\mathop {}\limits^4 } \right)\left( {\mathop {}\limits^5 } \right)\left( {\mathop {}\limits^6 } \right)\left( {\mathop {}\limits^7 } \right)$$
Now, 3 vowels can placed at any of the three places out of four marked 1, 3, 5, 7
Number of ways of arranging the vowels
\eqalign{ & = {}^4{P_3} \cr & = \left( {4 \times 3 \times 2} \right) \cr & = 24 \cr}
4 consonants at the remaining 4 positions may be arranged in $${}^4{P_4} = 4! =$$   24 ways
Required number of ways = (24 × 24) = 576
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