What are `""^(n)C_(0)` and `""^(n)C_(n)` ?

Answer

To solve the question "What are \( \binom{n}{0} \) and \( \binom{n}{n} \)?", we can break it down into two parts.

### Step 1: Calculate \( \binom{n}{0} \)

The notation \( \binom{n}{r} \) represents the number of ways to choose \( r \) objects from \( n \) objects. The formula for combinations is given by:

\[ \binom{n}{r} = \frac{n!}{(n-r)! \cdot r!} \]

For \( r = 0 \):

\[ \binom{n}{0} = \frac{n!}{(n-0)! \cdot 0!} = \frac{n!}{n! \cdot 0!} \]

Since \( 0! = 1 \):

\[ \binom{n}{0} = \frac{n!}{n! \cdot 1} = \frac{n!}{n!} = 1 \]

### Step 2: Calculate \( \binom{n}{n} \)

Now, we calculate \( \binom{n}{n} \):

Using the same formula for combinations:

\[ \binom{n}{n} = \frac{n!}{(n-n)! \cdot n!} = \frac{n!}{0! \cdot n!} \]

Again, since \( 0! = 1 \):

\[ \binom{n}{n} = \frac{n!}{1 \cdot n!} = \frac{n!}{n!} = 1 \]

### Final Answer

Thus, both \( \binom{n}{0} \) and \( \binom{n}{n} \) are equal to 1:

\[ \binom{n}{0} = 1 \quad \text{and} \quad \binom{n}{n} = 1 \]

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