In how many ways can the letters of the word STRANGE be arranfed so that the vowels may appear in the odd places ?

Options

A. A. 720
B. B. 1092
C. C. 1440
D. D. none of these

Answer

To solve the problem of arranging the letters of the word "STRANGE" such that the vowels appear in the odd positions, we can follow these steps:

### Step 1: Identify the letters and their types
The word "STRANGE" consists of 7 letters: S, T, R, A, N, G, E. Among these, the vowels are A and E, while the consonants are S, T, R, N, and G.

### Step 2: Identify the positions
In a 7-letter arrangement, the odd positions are 1, 3, 5, and 7. This gives us 4 odd positions available for the vowels.

### Step 3: Choose positions for the vowels
Since we have 2 vowels (A and E) and 4 available odd positions, we need to choose 2 out of these 4 positions for the vowels. The number of ways to choose 2 positions from 4 is given by the combination formula:
\[ \text{Number of ways to choose 2 positions} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Arrange the vowels in the chosen positions
After choosing the positions for the vowels, we can arrange the 2 vowels (A and E) in those positions. The number of ways to arrange 2 vowels is:
\[ 2! = 2 \]

### Step 5: Arrange the consonants
Now, we have 5 consonants (S, T, R, N, G) that need to fill the remaining 5 positions (which include the 2 odd positions not occupied by vowels and the 3 even positions). The number of ways to arrange these 5 consonants is:
\[ 5! = 120 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to choose the vowel positions, the arrangements of the vowels, and the arrangements of the consonants:
\[ \text{Total arrangements} = \binom{4}{2} \times 2! \times 5! = 6 \times 2 \times 120 = 1440 \]

Thus, the total number of ways to arrange the letters of the word "STRANGE" so that the vowels appear in the odd positions is **1440**.

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