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Difficulty: MediumCounting Principles

A communications center needs to assemble a rapid response team from a pool of 77 qualified technicians. The team must consist of exactly 11 team chief and 33 field technicians. The 33 field technicians share identical duties. How many different rapid response teams can be formed?

Answer: 140

Answer

140
To find the total number of possible teams, we must account for the distinct roles across two stages of selection. First, we select 11 team chief from the 77 available technicians, which gives 77 options. Next, we select 33 field technicians from the remaining 66 technicians. Because the field technicians have identical roles, the order in which they are selected does not matter. This requires calculating a combination: C(6,3)=6!3!(6−3)!=20C(6,3) = \frac{6!}{3!(6-3)!} = 20. Finally, using the Fundamental Counting Principle, we multiply the number of choices for the chief by the number of combinations for the field technicians: 7×20=1407 \times 20 = 140.

Step-by-Step Solution

1
Select the team chief.
7 possible choices
There are 7 available technicians, and 1 must be chosen for the distinct role of team chief.
2
Select the field technicians.
20 possible combinations
After choosing the chief, 6 technicians remain. We need to select 3 for identical roles. Since order does not matter, this is a combination: C(6,3) = (6 × 5 × 4) / (3 × 2 × 1) = 20.
3
Calculate the total number of unique teams.
140 teams
Using the Fundamental Counting Principle, multiply the independent choices for each stage: 7 × 20 = 140.

Key Concept

Combining the Fundamental Counting Principle with combinations for multi-stage selection.
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Topics

Counting Principles

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