Choose Teams
1. The problem asks: How many different teams of 3 can be chosen from a squad of 8?
2. This is a combination problem because the order of choosing team members does not matter.
3. The number of ways to choose $k$ items from $n$ items without regard to order is given by the combination formula:
$$ \binom{n}{k} = \frac{n!}{k! (n-k)!} $$
4. Here, $n=8$ and $k=3$. Plugging these into the formula:
$$ \binom{8}{3} = \frac{8!}{3! \times (8-3)!} = \frac{8!}{3! \times 5!} $$
5. Let's evaluate this step by step:
$$ 8! = 8 \times 7 \times 6 \times 5! $$
So,
$$ \binom{8}{3} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3!} $$
6. Calculate $3!$:
$$ 3! = 3 \times 2 \times 1 = 6 $$
7. Therefore:
$$ \binom{8}{3} = \frac{8 \times 7 \times 6}{6} $$
8. Simplify numerator and denominator:
$$ \frac{8 \times 7 \times 6}{6} = 8 \times 7 = 56 $$
9. Final Answer: There are 56 different teams of 3 that can be chosen from 8 people.