L.C.M. of 3!, 4! And 5! Is :

Options

A. 3!
B. 4!
C. 5!
D. 12!

Answer

To find the L.C.M. of \(3!\), \(4!\), and \(5!\), we can follow these steps:

### Step 1: Calculate the Factorials
First, we need to calculate the values of \(3!\), \(4!\), and \(5!\).

\[ 3! = 3 \times 2 \times 1 = 6 \]

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 2: Express Factorials in Terms of Each Other
Next, we can express \(4!\) and \(5!\) in terms of \(3!\):

\[ 4! = 4 \times 3! = 4 \times 6 = 24 \]

\[ 5! = 5 \times 4! = 5 \times 24 = 120 \]

### Step 3: Identify the Highest Factorial
To find the L.C.M., we look for the highest factorial among the three:

- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)

Since \(5!\) contains all the factors of \(4!\) and \(3!\), the L.C.M. will be \(5!\).

### Step 4: Conclusion
Thus, the L.C.M. of \(3!\), \(4!\), and \(5!\) is:

\[ \text{L.C.M.} = 5! = 120 \]

### Final Answer
The L.C.M. of \(3!\), \(4!\), and \(5!\) is \(120\).

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