Solve the problem using the idea of labeling.
-In how many ways can 11 volunteers be assigned to work in the governor's reelection campaign with 3 assigned to stuffing envelopes, 2 assigned to making phone calls, and 6 assigned to making signs?
A) 3,326,400
B) 1,108,800
C) 3,628,800
D) 4620

Question

Quizplus · Accepted Answer

This problem can be solved using the concept of combinations and permutations. Since the order of selection does not matter within each group but matters across the groups due to different tasks, we use combinations to select the groups and then multiply the results. First, choose 3 volunteers for stuffing envelopes from 11, which can be done in $ \binom{11}{3} $ ways. Then, from the remaining 8, choose 2 for making phone calls, which can be done in $ \binom{8}{2} $ ways. Finally, all 6 remaining will be making signs, but since this is the last group, there's only $ \binom{6}{6} = 1 $ way to choose them. The total number of ways to assign the volunteers is the product of these combinations:$$ \binom{11}{3} \times \binom{8}{2} \times \binom{6}{6} = \frac{11!}{3!8!} \times \frac{8!}{2!6!} \times 1 = \frac{11!}{3!2!6!} = 4620. $$This calculation simplifies to 4620 ways to assign the volunteers to the tasks.

Solve the Problem Using the Idea of Labeling