Let's work through each question step-by-step:

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Q1. Hockey Teams of 6 from 14 Players

To find how many teams of 6 players can be chosen from a group of 14 without considering the position, we use the combination formula: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ where $n = 14$ and $r = 6$.

$\binom{14}{6} = \frac{14!}{6!(14-6)!} = \frac{14!}{6! \times 8!}$ Calculating, we get: $\binom{14}{6} = 3003$

So, there are 3003 ways to choose the team.

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Q2. Selecting a 4-Person Bobsled Team

One of the team members must be from a group of 9 football players, and the other 3 can be from a separate group of 8 athletes.

1. Choose 1 person from 9 football players: $\binom{9}{1} = 9$

2. Choose 3 people from 8 other athletes: $\binom{8}{3} = \frac{8!}{3! \times (8-3)!} = \frac{8!}{3! \times 5!} = 56$

Now, multiply the two results: $9 \times 56 = 504$

So, there are 504 ways to select the team.

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Q3. Hiring 5 Flight Attendants from 8 Applicants

To form a group of 5 attendants from 8 applicants, we calculate the combination: $\binom{8}{5} = \frac{8!}{5! \times (8-5)!} = \frac{8!}{5! \times 3!} = 56$

Thus, there are 56 possible groups of 5 attendants.

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Q4. Forming a Committee of 5 Men and 6 Women from 10 Men and 12 Women

1. Choose 5 men from 10: $\binom{10}{5} = \frac{10!}{5! \times (10-5)!} = 252$

2. Choose 6 women from 12: $\binom{12}{6} = \frac{12!}{6! \times (12-6)!} = 924$

Multiply the results: $252 \times 924 = 232,848$

There are 232,848 ways to form this committee.

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Q5. Selecting 1st, 2nd, and 3rd Places from 8 Students

Since order matters, this is a permutation problem: $P(n, r) = \frac{n!}{(n-r)!}$ where $n = 8$ and $r = 3$: $P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 336$

There are 336 ways to select the winners.

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Q6. Choosing 3 Different Fruits for 3 People from 5 Fruits

Since each person chooses one different fruit, this is a permutation problem where order matters: $P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 60$

There are 60 ways for the three people to choose different fruits.

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Let me know if you have any questions or need further clarification on any step! Here are some additional practice questions to expand your understanding:

1. How would the calculation change in Q1 if the team size was 7?
2. For Q2, what if 2 people must come from the football group and the remaining 2 from the other athletes?
3. What would be the result for Q3 if there were only 7 applicants instead of 8?
4. In Q4, how many committees could be formed if 4 men and 5 women were required instead?
5. For Q5, what if only the 1st and 2nd places needed to be chosen?

Tip: Remember, permutations are used when order matters, and combinations are used when it does not.