Is `3!+4!=7!` ?

Answer

To determine if \(3! + 4! = 7!\), we will calculate both sides of the equation step by step.

### Step 1: Calculate the Left-Hand Side (LHS)
The LHS is \(3! + 4!\).

#### Step 1.1: Calculate \(3!\)
The factorial of 3, denoted as \(3!\), is calculated as:
\[ 3! = 3 \times 2 \times 1 = 6 \]

#### Step 1.2: Calculate \(4!\)
The factorial of 4, denoted as \(4!\), is calculated as:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

#### Step 1.3: Add \(3!\) and \(4!\)
Now we add the two results:
\[ 3! + 4! = 6 + 24 = 30 \]

### Step 2: Calculate the Right-Hand Side (RHS)
The RHS is \(7!\).

#### Step 2.1: Calculate \(7!\)
The factorial of 7, denoted as \(7!\), is calculated as:
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]
Calculating this step by step:
- \(7 \times 6 = 42\)
- \(42 \times 5 = 210\)
- \(210 \times 4 = 840\)
- \(840 \times 3 = 2520\)
- \(2520 \times 2 = 5040\)
- \(5040 \times 1 = 5040\)

So, \(7! = 5040\).

### Step 3: Compare LHS and RHS
Now we compare the two sides:
- LHS = \(30\)
- RHS = \(5040\)

Since \(30 \neq 5040\), we conclude that:
\[ 3! + 4! \neq 7! \]

### Final Answer
Thus, \(3! + 4! \) is not equal to \(7!\).

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