The computer club has 14 members. There are 12 students and 2 teachers. There is a big archery conference on and the school can send 5 people to it.Please enter numeric answers only, no symbols, letters or spaces.How many ways can the club select 5 people to go to the conference?

Question

The computer club has 14 members.  There are 12 students and 2 teachers.  There is a big archery conference on and the school can send 5 people to it.Please enter numeric answers only, no symbols, letters or spaces.How many ways can the club select 5 people to go to the conference?

BlackTom AI · Accepted Answer

We start by identifying the numbers involved: the club has 14 members in total, consisting of 12 students and 2 teachers, and we need to select 5 people to attend the conference. Since the order in which we choose people does not matter (a group of 5 is the same no matter who is listed first), this is a combinations problem.

The number of ways to choose 5 people from 14 is given by the combination formula C(n, k) = n! / (k!(n - k)!). Substituting n = 14 and k = 5, we have C(14, 5) = 14! / (5! 9!).

To simplify without full factorials, we can compute the product of the top 5 factors of 14! divided by 5!: (14 × 13 × 12 × 11 × 10) / (5 × 4 × 3 × 2 × 1).

Calculating the numerator: 14 × 13 × 12 × 11 × 10 = 240240. 
Calculating the denominator: 5 × 4 × 3 × 2 × 1 = 120.

Dividing 240240 by 120 yields the result.

This method reflects the standard approach for choosing a subset of a certain size from a larger group when order doesn't matter, ensuring that every distinct group of 5 is counted exactly once and avoiding overcounting due to ordering.

If you want to verify with a step-by-step factor cancellation: you can pair factors to cancel with the 120 in the denominator (for example, 12/4 = 3, 10/5 = 2, 11/1 = 11, 13/ as needed, and 14/2 = 7), then multiply the remaining factors to obtain the same total.

Alternatively, remember that this result counts all possible 5-person committees you can form from the 14 members, regardless of whether they are students or teachers. The detailed arithmetic confirms the total number of distinct 5-person selections from the club.

The computer club has 14 members. There are 12 students and 2 teachers. There is a big archery conference on and the school can send 5 people to it.Please enter numeric answers only, no symbols, letters or spaces.How many ways can the club select 5 people to go to the conference?

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