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

### Concept & Positional Permutations When specific items are restricted to specific positions, calculate the permutations for those constrained items first, then arrange the remaining items in the leftover positions. Formula for arranging $r$ items in $n$ available slots: $^nP_r = \frac{n!}{(n-r)!}$ ### Step-by-Step Solution * **Analyze the Word:** "MACHINE" contains 7 letters. * **Identify Vowels and Consonants:** * Vowels: A, I, E (3 letters) * Consonants: M, C, H, N (4 letters) * **Identify Odd Positions:** In a 7-letter word, the odd positions are 1st, 3rd, 5th, and 7th. There are 4 odd positions. * **Arrange Vowels:** The 3 vowels must be placed in the 4 available odd positions. * Number of ways = $^4P_3 = 4 \times 3 \times 2 = 24$ ways. * **Arrange Consonants:** After placing the 3 vowels, there is 1 odd position left, plus the 3 even positions (2nd, 4th, 6th). This gives a total of 4 remaining positions for the 4 consonants. * Number of ways = $4! = 24$ ways. * **Calculate Total Arrangements:** Multiply the independent arrangements. * Total ways = $24 \times 24 = 576$. ### Exam Strategy & Shortcut Deconstruct the problem into independent tasks: 3 vowels into 4 slots is $4 \times 3 \times 2 = 24$. The remaining 4 letters into the remaining 4 slots is $4! = 24$. Mentally calculate $24^2 = 576$. Memorizing squares up to $25$ saves significant time here. ### Common Pitfall A frequent error is assuming the consonants must only go into the even positions. The problem states vowels must occupy *only* odd positions, but it doesn't restrict consonants from occupying the leftover odd position. ### Final Answer Therefore, the correct answer is **576**.