Five persons are to be seated in a 10 seater suv In how many ways can they be seated

There are three parts to this. There's always a driver and all 4 can drive so whatever the combination in the other 5 seats, you are going to multiply that combination by 4 to represent the 4 different sets of people who could be in the other seats.

Next, you have three people to fill five seats. You will want to determine the number of combinations of filled seats and the order of the 3 people as separate factors. If seats 1 & 2 are always occupied, you have 3 combinations. If seats 1 & 3 are always occupied, you have 2 new additional combinations. If seats 1 & 4 are occupied, there is 1 new combination. So if seat 1 is always occupied, there are 6 combinations. Now, if seat 1 is always vacant, you are distributing through seats 2 through 5. Same method of counting shows 3 new combinations with seat 1 vacant and seat 2 occupied. Finally, if you vacate seats 1 & 2, you have one new combination. So, that's 10 combinations of occupied seats. Then there is the order of each person sitting for any particular combination of occupied seats. That's a simple 3x2. So 4 (driver sets) x 10 (seating combinations) x 6 (order of people) = 240 combinations.

Now, we want more mathematical eloquence than just counting the five seat positions. If we were dealing with a genome instead of a 6 passenger car, counting wouldn't be very practical. To count the combination of occupied seats, I did that counting occupied seats. I could have counted vacant combinations. If seat 1 is vacant there are 4 combinations for the other vacant seat. If seat 2 is always vacant there are 3 new combinations, seat 3 vacant adds 2 new combinations and seat 4 one new combination. That still adds up to 10, but it is still not eloquent. If we are using Google sheets, we could use combin(5,2) to get the number of combinations, so what is the math behind combin? See, https://corporatefinanceinstitute.com/resources/excel/functions/combin-function/ for the actual formula.

Please point out my errors and be critical of the answer. I'm old and have forgotten a lot.

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