Question

In how many ways can you choose 2 people out of a group of 10 people?

Answer 1

To determine how many ways we can choose 2 people out of a group of 10, we can use the combination formula, which is given by:

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

where \( n \) is the total number of people, \( r \) is the number of people to choose, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.

In this case, \( n = 10 \) and \( r = 2 \). Thus, we calculate:

\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!}
\]

Next, we simplify the factorials:

\[
10! = 10 \times 9 \times 8!
\]

So,

\[
\binom{10}{2} = \frac{10 \times 9 \times 8!}{2! \times 8!}
\]

The \( 8! \) in the numerator and denominator cancels out:

\[
\binom{10}{2} = \frac{10 \times 9}{2!}
\]

Now we compute \( 2! \):

\[
2! = 2 \times 1 = 2
\]

Thus, the expression becomes:

\[
\binom{10}{2} = \frac{10 \times 9}{2} = \frac{90}{2} = 45
\]

Therefore, the number of ways to choose 2 people out of a group of 10 is:

\[
\boxed{45}
\]