Building a Diverse Team from a Newsroom: Combinatorial Approach

When tasked with building a team of 7 workers from a pool of 20 reporters, 15 graphic designers, 10 photographers, and 10 editors, we need to ensure that the team includes at least one worker from each role. This problem can be tackled using combinatorial math, specifically the binomial coefficient, also known as the number of combinations. Let's explore this in detail.

Step-by-Step Solution

We start by ensuring that the team includes at least one worker from each role: reporter, graphic designer, photographer, and editor. Here's how we do it:

Assign one reporter, one graphic designer, one photographer, and one editor to the team. This step can be completed in:

Number of ways to assign one worker from each role:

20 (reporters) times; 15 (graphic designers) times; 10 (photographers) times; 10 (editors)

20 * 15 * 10 * 10 30,000 After assigning one worker from each role, we have 51 workers remaining (55 total minus 4 already assigned). Now, we need to choose 3 more workers out of these 51. The number of combinations is:

Number of ways to choose 3 workers out of 51:

binom{51}{3} frac{51!}{3!(51-3)!} frac{51!}{3!48!}

frac{51 times 50 times 49}{3 times 2 times 1} 20825

Therefore, the total number of ways to build a team of 7 workers from the newsroom, ensuring at least one worker from each role, is:

Total number of ways:

30,000 times; 20,825 624,750,000

Combinatorial Formulation

We can also express the problem using combinatorial notation. Let's define:

R: Number of ways to form a team without at least one reporter G: Number of ways to form a team without at least one graphic designer P: Number of ways to form a team without at least one photographer E: Number of ways to form a team without at least one editor

The required number of ways is given by:

Total number of ways when disregarding conditions:

(binom{55}{7})

Number of ways to form a team without at least one reporter (R):

(binom{55-20}{7} binom{35}{7})

Similarly for graphic designers, photographers, and editors:

(binom{55-15}{7} binom{40}{7})

Using Inclusion-Exclusion Principle:

(binom{55}{7} - (R cup G cup P cup E))

(binom{55}{7} - (R G P E - (R cap G) - (R cap P) - (R cap E) - (G cap P) - (G cap E) - (P cap E) (R cap G cap P) (R cap G cap E) (R cap P cap E) (G cap P cap E) - (R cap G cap P cap E)))

Conclusion

Using combinatorial methods, we can effectively determine the number of ways to build a diverse team from a newsroom, ensuring that it includes at least one worker from each role. This approach not only satisfies the conditions but also showcases the power of combinatorics in solving complex problems.