Calculate 8 choose 1 in combinatorics.

In combinatorics, the expression ''8 choose 1'' refers to the number of ways to select 1 item from a set of 8 distinct items. This is a common problem in the study of combinations, which are selections where the order of the items does not matter.

To calculate ''8 choose 1'', we use the binomial coefficient formula, which is given by:

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

where \( n \) is the total number of items to choose from, \( k \) is the number of items to choose, and \( ! \) denotes factorial, which is the product of all positive integers up to that number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Let's apply this formula to our problem:

\[ \binom{8}{1} = \frac{8!}{1!(8-1)!} \]

First, we calculate the factorials:

\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]
\[ 1! = 1 \]
\[ (8-1)! = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Now, we can simplify the expression by canceling out the common terms in the numerator and the denominator:

\[ \binom{8}{1} = \frac{8 \times 7!}{1 \times 7!} \]

Since \( 7! \) appears in both the numerator and the denominator, they cancel each other out:

\[ \binom{8}{1} = \frac{8}{1} \]

Finally, we are left with:

\[ \binom{8}{1} = 8 \]

Therefore, there are 8 ways to choose 1 item from a set of 8 distinct items. This result makes intuitive sense because for each item in the set, there is exactly one way to choose it, resulting in 8 possible selections.