If the Math Olympiad Club consists of 14 students, how many different teams of 6 students can be formed for competitions?

To solve this problem, we need to use the concept of combinations in mathematics. A combination is a selection of items without considering the order.

The formula for combinations is:

C(n, k) = n! / [k!(n-k)!]

where:

• n is the total number of items,
• k is the number of items to choose,
• "!" denotes factorial, which means multiplying all positive integers up to that number.

In this case, n = 14 (the total number of students) and k = 6 (the number of students in a team).

So, we can substitute these values into the formula:

C(14, 6) = 14! / [6!(14-6)!]

Calculating the factorials:

14! = 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 6! = 6 × 5 × 4 × 3 × 2 × 1 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Substituting these values back into the formula:

C(14, 6) = 14! / [6! × 8!]

After calculating the above expression, we find that there are 3003 different teams of 6 students that can be formed from a club of 14 students.