There are 20 seniors serving the student council of the Cebu Institute of Technology this year. Of these, 3 have not served before, 10 served on the council in their junior years, 9 in their sophomore years, and 11 in their freshman years. There are 5 who served during both their sophomore and junior years, 6 during both their freshman and junior years, and 4 during both their freshman and sophomore years. How many seniors served on the student council during each of the four years in high school?
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Answer:15 seniors served on the student council during their freshman year, 14 seniors served during their sophomore year, 16 seniors served during their junior year, and 3 seniors have never served before.
Step-by-step explanation:
Final answer:
Using inclusion and exclusion principles, we find that 2 seniors served on the student council during each of the four years in high school.
Explanation:
The problem can be solved using the Principle of Inclusion and Exclusion (PIE), a common technique in combinatorial mathematics. First, we add the number of seniors serving in their freshman, sophomore, and junior years: 3 (never served) + 10 (junior) + 9 (sophomore) + 11 (freshman) giving us 33.
Then, we subtract the number of seniors who served during both sophomore and junior years, freshman and junior years, and freshman and sophomore years: 33 - 5 (sophomore and junior) - 6 (freshman and junior) - 4 (freshman and sophomore). This results in 18.
However, from the initial condition we know that there are 20 seniors in total. Therefore, the two 'extra' seniors must have served all four years in high school. Thus we find that 2 seniors served on the student council during each of the four years in high school.
Learn more about Inclusion and Exclusion Principle here:
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