Sports of the 25 baseball players on the Baltimore Orioles’ 2005 roster, 17 threw right-handed, 12 were over 30 years old, and 9 both threw right-handed and were over 30 years old. How many players on the team neither threw right-handed nor were over 30 years old?
The Correct Answer and Explanation is:
We are given:
- Total players = 25
- Number who threw right-handed = 17
- Number who were over 30 years old = 12
- Number who both threw right-handed and were over 30 = 9
We are asked to find how many players neither threw right-handed nor were over 30 years old.
This is a classic problem involving sets and Venn diagrams.
Let:
- RRR be the set of players who threw right-handed
- OOO be the set of players who were over 30 years old
- R∩OR \cap OR∩O is the set of players who are in both sets (right-handed and over 30)
We can use the principle of inclusion and exclusion to find how many players were in either RRR or OOO (or both):∣R∪O∣=∣R∣+∣O∣−∣R∩O∣∣R∪O∣=17+12−9=20|R \cup O| = |R| + |O| – |R \cap O| \\ |R \cup O| = 17 + 12 – 9 = 20∣R∪O∣=∣R∣+∣O∣−∣R∩O∣∣R∪O∣=17+12−9=20
This tells us that 20 players were either right-handed, over 30, or both.
Since the total number of players is 25, we subtract:25−20=525 – 20 = 525−20=5
So, 5 players were neither right-handed nor over 30 years old.
Final Answer:
5\boxed{5}5
Explanation:
This type of problem is best solved using set theory. You are dealing with overlapping groups and must account for double-counting. Adding the two categories gives 17 + 12 = 29, but 9 players were counted twice because they belong to both groups. Subtracting the overlap gives the correct total for those in either group. The remainder are those in neither group. This technique is often used in surveys and data classification tasks to ensure accurate counts when categories overlap.
