Let the number of people who chose Package A be x and the number of people who chose Package B be y.

Let the number of extra nights be z.

According to the problem statement, we have the following information:

1. The total cost of the trip is Rs. 1800: 500x + 300y + 50z = 1800.

2. The total number of people is 4: x + y = 4.

Substituting the value of y from equation 2 into equation 1,

we get: 500x + 300(4 - x) + 50z = 1800.

Solving for z, we get

$`z = \frac {1800 – 500x – 1200 + 300x}{50} \\ =\frac{600 – 200x}{50} `.$

Since z must be a whole number, we can conclude that x must be divisible by 5.

The possible values of x are 0, 5, 10, ..., 50.

However, since the total number of people is 4, we can conclude that the maximum possible value of x is 4.

Therefore, the only possible value of x is 0.

Substituting this value of x into equation 2,

we get: y = 4 - x = 4.

Substituting these values of x and y into equation 1,

we get: 500(0) + 300(4) + 50z = 1800.

Solving for z, we get $z =\frac{1800 – 1200}{50} = 12.$

Therefore, the group stayed for an extra 12 nights.

Therefore, the correct option is (E).

Absolute Value