Players: 25 offense, 28 defense, 17 special teams. Overlaps: 7 both offense/defense, 3 both offense/special, 13 both defense/special. No player is in all three. How many players are on the team?

Rationale

There are 47 players on the team.

To find the total number of players, we can apply the principle of inclusion-exclusion. We start with the total number of players in each category and then subtract the overlaps, while ensuring that we don't double-count any players who belong to multiple categories.

A (47) CORRECT

This answer correctly accounts for the total number of unique players by applying the inclusion-exclusion principle. We add the number of players in all categories (25 offense + 28 defense + 17 special teams) and subtract the overlaps: 7 who play both offense and defense, 3 who play both offense and special teams, and 13 who play both defense and special teams. This gives us a total of 47 unique players.

B (48) YOUR ANSWER

This option incorrectly adds one extra player by failing to properly account for the overlaps. The inclusion-exclusion method ensures that no player is counted more than once. Thus, 48 is not supported by the calculated total.

C (49) YOUR ANSWER

Choosing 49 also miscalculates the total by including too many players. Similar to option B, this answer overlooks the overlaps of players between categories, leading to an inflated count.

D (50) YOUR ANSWER

This choice suggests an even higher total, which fails to recognize the correct number of overlaps. It does not reflect the accurate outcome derived from the inclusion-exclusion principle.

E (51) YOUR ANSWER

This option indicates a total that is significantly over the actual count. It does not take into account the overlaps properly, leading to an erroneous total that does not align with the calculated result.

Conclusion

The correct number of players on the team is 47, derived from the inclusion-exclusion principle applied to the three categories of players and their respective overlaps. By accurately subtracting the overlaps, we ensure that each player is counted only once, which is essential for determining the true unique player count in this scenario.

Players: 25 offense, 28 defense, 17 special teams. Overlaps: 7 both offense/defense, 3 both offense/special, 13 both defense/special. No player is in all three. How many players are on the team?

There are 47 players on the team.

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