## Combinatorial analysis

THE PROBLEM RESOLVES AS FOLLOWS:

There are 7 people, and one can never go in a front seat.

Let's call this person John, for example.

So first let's calculate the number of ways to fill the car WITHOUT John, using only the other six people:

As we have 6 people and 5 seats in the car then we calculate the arrangement of 6 elements, taken 5 to 5:

THE_{6,5}= 720

Now let's calculate the number of ways to fill the car with John.

We know John can't be in the front seats, so he must be in one of the three back seats.

So we fixed John in one of the rear seats (so there are 4 seats left in the car), and then we calculated the number of ways to put the other 6 people in those 4 seats, that is, an arrangement of 6 elements, taken 4 to 4:

THE_{6,4}= 360

John can be in any of the three back seats, so we should multiply this result by 3:

3 x A_{6,4}= 3x360 = 1080

The total number of ways to fill the car is the sum of the two arrangements (WITH John and NO John).

So total number is 720 + 1080 = 1800 ways !!!

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