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n = 2p - 2

To represent the total number of soccer players, n, in terms of the seating capacity, p, we can analyze the seating arrangement. One van is full, accommodating p players, while the second van has 2 empty seats, meaning it can hold p - 2 players. Therefore, the equation becomes n = p + (p - 2), simplifying to n = 2p - 2.

A) n = 2p + 2

This option suggests that the total number of players exceeds the total seating capacity by 2, which is not consistent with the scenario presented. Since one van is full and the other has 2 empty seats, this equation does not accurately reflect the situation.

B) n = 2p - 2

This is the correct representation of n in terms of p. The first van can hold p players, and the second van has 2 empty seats, thus it holds p - 2 players. Combining these gives n = p + (p - 2) = 2p - 2, which accurately describes the total number of players.

C) n = 2(p - 2)

This option implies that the total number of players is double the capacity of the second van after accounting for the empty seats. However, it does not correctly represent the configuration of the two vans, as it does not consider the total seating capacity adequately.

D) n = p + 2

This option indicates that the total number of players is equal to the capacity of one van plus 2 additional players. This does not align with the information given, as it fails to account for the fact that one van is full and the other has 2 empty seats.

Conclusion

The correct answer, n = 2p - 2, accurately reflects the seating arrangement of the two vans given in the problem. The other options either miscalculate or misinterpret the seating capacities, failing to consider the configuration of the players in relation to the empty seats. Thus, B is the only option that correctly represents the scenario described.