The number of permutations of the letters of the word LUMINARY such that neither the pattern LURY nor MINA occurs is :

Options

A. 46800
B. 24600
C. 40086
D. none of these

Answer

To solve the problem of finding the number of permutations of the letters in the word "LUMINARY" such that neither the pattern "LURY" nor "MINA" occurs, we can follow these steps:

### Step 1: Calculate the total number of permutations of the letters in "LUMINARY".
The word "LUMINARY" consists of 8 distinct letters. The total number of permutations of these letters can be calculated using the factorial of the number of letters.

\[ \text{Total permutations} = 8! = 40320 \]

### Step 2: Calculate the permutations where "LURY" occurs as a block.
When "LURY" occurs together, we can treat it as a single unit or block. Thus, we have the following blocks to arrange: "LURY", "M", "I", "N", "A", "R", "Y". This gives us a total of 5 units to arrange.

The number of permutations of these 5 units is:

\[ \text{Permutations with LURY} = 5! = 120 \]

### Step 3: Calculate the permutations where "MINA" occurs as a block.
Similarly, when "MINA" occurs together, we treat it as a single block. The blocks to arrange are: "MINA", "L", "U", "R", "Y". Again, we have 5 units to arrange.

The number of permutations of these 5 units is:

\[ \text{Permutations with MINA} = 5! = 120 \]

### Step 4: Calculate the permutations where both "LURY" and "MINA" occur as blocks.
Now, we treat both "LURY" and "MINA" as single blocks. The blocks to arrange are: "LURY", "MINA", "R", "Y". This gives us a total of 4 units to arrange.

The number of permutations of these 4 units is:

\[ \text{Permutations with both LURY and MINA} = 4! = 24 \]

### Step 5: Apply the principle of inclusion-exclusion.
To find the total number of permutations where either "LURY" or "MINA" occurs, we can use the principle of inclusion-exclusion:

\[ \text{Permutations with LURY or MINA} = (\text{Permutations with LURY}) + (\text{Permutations with MINA}) - (\text{Permutations with both}) \]

Substituting the values we found:

\[ \text{Permutations with LURY or MINA} = 120 + 120 - 24 = 216 \]

### Step 6: Calculate the permutations where neither "LURY" nor "MINA" occurs.
Finally, we subtract the number of permutations where either "LURY" or "MINA" occurs from the total permutations:

\[ \text{Permutations with neither LURY nor MINA} = \text{Total permutations} - \text{Permutations with LURY or MINA} \]

Substituting the values:

\[ \text{Permutations with neither LURY nor MINA} = 40320 - 216 = 40104 \]

### Final Answer
The number of permutations of the letters of the word "LUMINARY" such that neither the pattern "LURY" nor "MINA" occurs is **40104**.