How many words can be formed with the letters of the word 'LUCKNOW' in which L,U,C occupy odd positions?

Answer

To solve the problem of how many words can be formed with the letters of the word 'LUCKNOW' in which L, U, and C occupy odd positions, we can follow these steps:

### Step 1: Identify the Positions
The word 'LUCKNOW' has 7 letters. The positions can be labeled as follows:
1. Position 1 (odd)
2. Position 2 (even)
3. Position 3 (odd)
4. Position 4 (even)
5. Position 5 (odd)
6. Position 6 (even)
7. Position 7 (odd)

The odd positions available are 1, 3, 5, and 7.

### Step 2: Determine the Letters for Odd Positions
We need to place the letters L, U, and C in the odd positions. Since there are 4 odd positions and only 3 letters (L, U, C), we will need to select one additional letter from the remaining letters of the word 'LUCKNOW'.

The remaining letters after L, U, and C are K, N, O, and W. We can choose one letter from these 4 letters to fill the fourth odd position.

### Step 3: Choose the Additional Letter
The number of ways to choose 1 letter from the 4 remaining letters (K, N, O, W) is given by:
\[ \binom{4}{1} = 4 \]

### Step 4: Arrange the Letters in Odd Positions
Now we have 4 letters to arrange in the odd positions: L, U, C, and the chosen letter (let's call it α). The number of ways to arrange these 4 letters is:
\[ 4! = 24 \]

### Step 5: Arrange the Remaining Letters in Even Positions
After placing L, U, C, and α in the odd positions, we have 3 letters left (the remaining letters from 'LUCKNOW' that were not chosen). These letters will occupy the 3 even positions. The number of ways to arrange these 3 letters is:
\[ 3! = 6 \]

### Step 6: Calculate the Total Arrangements
To find the total number of arrangements, we multiply the number of ways to choose the additional letter, the arrangements of the letters in odd positions, and the arrangements of the letters in even positions:
\[ \text{Total arrangements} = \binom{4}{1} \times 4! \times 3! = 4 \times 24 \times 6 \]

Calculating this gives:
\[ 4 \times 24 = 96 \]
\[ 96 \times 6 = 576 \]

### Final Answer
Thus, the total number of words that can be formed with the letters of the word 'LUCKNOW' in which L, U, and C occupy odd positions is **576**.

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