Explam why $$ 0!=1 $$ and $$ 1!=1 $$

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Great question! Understanding why $$ 0! = 1 $$ and $$ 1! = 1 $$ involves delving into the definition of factorials and their applications in mathematics, especially in combinatorics and algebra. Let's break it down:

### **Factorial Definition**

The factorial of a non-negative integer $$ n $$, denoted by $$ n! $$, is defined as the product of all positive integers from 1 up to $$ n $$. Mathematically,

$$
n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1
$$

For example:
- $$ 3! = 3 \times 2 \times 1 = 6 $$
- $$ 2! = 2 \times 1 = 2 $$

### **Why $$ 1! = 1 $$**

Applying the definition directly:

$$
1! = 1
$$

This is straightforward because there's only one positive integer (which is 1) to multiply.

### **Why $$ 0! = 1 $$**

At first glance, it might seem odd to define $$ 0! $$ as 1, since there are no positive integers to multiply. However, there are several compelling reasons for this definition:

1. **Empty Product Convention**:
   - In mathematics, the product of no numbers at all is defined as 1. This concept is known as the "empty product."
   - Since $$ 0! $$ involves multiplying no numbers, it follows that $$ 0! = 1 $$.

2. **Combinatorial Interpretation**:
   - Factorials are closely related to combinations and permutations. Specifically, $$ n! $$ represents the number of ways to arrange $$ n $$ distinct objects.
   - How many ways are there to arrange zero objects? There's exactly **one** way: do nothing.
   - Therefore, $$ 0! = 1 $$.

3. **Recurrence Relation for Factorials**:
   - Factorials satisfy the recurrence relation:
     $$
     n! = n \times (n-1)!
     $$
   - Let's apply this when $$ n = 1 $$:
     $$
     1! = 1 \times 0!
     $$
     Since $$ 1! $$ is known to be 1, we can solve for $$ 0! $$:
     $$
     1 = 1 \times 0! \implies 0! = 1
     $$

4. **Mathematical Consistency**:
   - Defining $$ 0! = 1 $$ ensures consistency across various mathematical formulas and functions, such as the binomial coefficient:
     $$
     \binom{n}{k} = \frac{n!}{k!(n-k)!}
     $$
     For example, when $$ k = 0 $$ or $$ k = n $$, this formula simplifies correctly only if $$ 0! = 1 $$.

### **Summary**

- **$$ 1! = 1 $$**: Directly from the definition since there's only one positive integer to multiply.
- **$$ 0! = 1 $$**: Established by the empty product convention, combinatorial logic (one way to arrange nothing), the factorial recurrence relation, and to maintain consistency in mathematical formulas.

Understanding these foundations helps ensure that factorial-related formulas and equations work seamlessly across all non-negative integers.
$$ 0! = 1 $$ because there's one way to arrange zero objects, and $$ 1! = 1 $$ by the definition of factorial.

Explam why \( 0!=1 \) and \( 1!=1 \)

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