Seven friends are riding the bus to school:

- Cha and Bai are on 2 different buses.
- Bai, Abu and Don are on 3 different buses.
- Don, Gia and Fan are on 3 different buses.
- Abu, Eva and Bai are on 3 different buses.
- Gia and Eva are on 2 different buses.
- Fan, Cha and Gia are on 3 different buses.
- Cha and Eva are on 2 different buses.

What is the least possible number of buses on which the friends could be riding?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Here is one way in which the seven friends can ride on four buses so that the seven restrictions are satisfied:

| Bus 1 | Bus 2 | Bus 3 | Bus 4 |
| :---: | :---: | :---: | :---: |
| Abu | Bai | Don | Gia |
| Cha | Fan | Eva |  |

At least 3 buses are needed because of the groups of 3 friends who must all be on different buses.

We will now show that it is impossible for the 7 friends to travel on only 3 buses.

Suppose that the seven friends could be put on 3 buses.

Since Abu, Bai and Don are on 3 different buses, then we assign them to three buses that we can call Bus 1, Bus 2 and Bus 3, respectively. (See the table below.)

Since Abu, Bai and Eva are on 3 different buses, then Eva must be on Bus 3.

Since Cha and Bai are on 2 different buses and Cha and Eva are on 2 different buses, then Cha cannot be on Bus 2 or Bus 3, so Cha is on Bus 1 .

So far, this gives

| Bus 1 | Bus 2 | Bus 3 |
| :---: | :---: | :---: |
| Abu | Bai | Don |
| Cha |  | Eva |

The remaining two friends are Fan and Gia.

Since Fan, Cha and Gia are on 3 different buses, then neither Fan nor Gia is on Bus 1.

Since Don, Gia and Fan are on 3 different buses, then neither Fan nor Gia is on Bus 3.

Since Gia and Fan are on separate buses, they cannot both be on Bus 2, which means that the seven friends cannot be on 3 buses only.

Therefore, the minimum number of buses needed is 4 .

ANswer: (B)