Question

Student Painters, which specializes in painting the exterior of residential buildings, has eleven people available to be organized into six-person and five-person teams.

(a) In how many ways can the six-person team be formed? ways

(b) In how many ways can the five-person team be formed? ways

(c) In how many ways can the company organize the available people into six- or five-person teams?

Answer 1

To calculate the number of ways to form teams, the combination formula is used. There are 462 ways to form both a six-person and a five-person team from 11 people. Since choosing one team automatically defines the other, there are 462 ways to organize into six- or five-person teams.

Step-by-step explanation:

The question involves calculating the number of ways to form teams from a set number of individuals, which is a problem that can be solved using combinations in mathematics. Specifically, we'll be using the combination formula nCr = n! / (r!(n - r)!) where n is the total number of items, r is the number of items to choose, and ! denotes factorial.

Part (a)

To form a six-person team from 11 individuals, we use the formula 11C6:

11C6 = 11! / (6!(11 - 6)!) = 11! / (6!5!) = 462

There are 462 ways to form a six-person team.

Part (b)

To form a five-person team, we use the formula 11C5:

11C5 = 11! / (5!(11 - 5)!) = 11! / (5!6!) = 462

Interestingly, there are also 462 ways to form a five-person team.

Part (c)

Since forming a six-person team automatically leaves the remaining five people as another team, we do not need to calculate them separately for the combination. Thus, the number of ways to organize the individuals into either a six- or five-person team is equal to the ways of forming just the six-person team.

Therefore, the company can organize the available people into six-person or five-person teams in 462 ways.