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Q2:
Is $3! + 4! = 7!$?
Solution :
Step 1: Definition of Factorial Notation
The factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. The definition is given as: $n! = n \times (n-1) \times (n-2) \times \dots \times 1$ [By the fundamental definition of factorials in combinatorics].
Step 2: Calculation of Left-Hand Side (LHS)
The expression on the left-hand side is $3! + 4!$.
- Calculate $3!$: $3! = 3 \times 2 \times 1 = 6$
- Calculate $4!$: $4! = 4 \times 3 \times 2 \times 1 = 24$
Summing these values: $3! + 4! = 6 + 24 = 30$ [Arithmetic addition].
Step 3: Calculation of Right-Hand Side (RHS)
The expression on the right-hand side is $7!$.
$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Performing the multiplication sequentially: $7 \times 6 = 42$ $42 \times 5 = 210$ $210 \times 4 = 840$ $840 \times 3 = 2520$ $2520 \times 2 = 5040$ $5040 \times 1 = 5040$ [Arithmetic multiplication].
Step 4: Comparison and Analysis
| Expression | Result |
|---|---|
| $LHS (3! + 4!)$ | $30$ |
| $RHS (7!)$ | $5040$ |
Comparing the calculated values: $30 \neq 5040$.
Step 5: Theoretical Generalization
In general, for any positive integers $m$ and $n$ where $m, n > 1$, $(m! + n!) \neq (m+n)!$ because the factorial function grows super-exponentially. Specifically, by the properties of factorials: $(m+n)! = (m+n) \times (m+n-1) \times \dots \times (n+1) \times n!$ This product is significantly larger than the sum of two individual factorials.
Final Solution: No, $3! + 4! \neq 7!$ because $30 \neq 5040$.
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