My school's math club has 6 boys and 8 girls.  I need to select a team to send to the state math competition.  We want 6 people on the team.  In how many ways can I select the team without restrictions?
With no restrictions, we are merely picking 6 students out of 14.  This is $\binom{14}{6} = \boxed{3003}$.
3003
Level 2
Counting & Probability
Problem: My school's math club has 6 boys and 8 girls.  I need to select a team to send to the state math competition.  We want 6 people on the team.  In how many ways can I select the team without restrictions?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of ways to select a team of 6 people from a total of 14 students (6 boys and 8 girls), we need to use the concept of combinations. The formula for combinations is given by C(n, k) = 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.

In this scenario, n is 14 (the total number of students) and k is 6 (the number of students to select). We can calculate C(14, 6) as follows:

1. Calculate the factorial of 14: 14! = 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
2. Calculate the factorial of 6: 6! = 6 × 5 × 4 × 3 × 2 × 1.
3. Calculate the factorial of (14 - 6), which is 8: 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
4. Substitute these values into the combination formula: C(14, 6) = 14! / (6! × 8!).
5. Simplify the expression by canceling out the common factorials in the numerator and denominator.

After performing the calculations, we find that C(14, 6) equals 3003. Thus, the total number of ways to select the team is 3003.<|END_OF_LATENT|> The final answer is $3003$.