Henry Ernest Dudeney/Puzzles and Curious Problems/288 - Seating the Party/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $288$

On a family outing, Dora asked in how many ways they could all be seated.

There were $6$ of them, three of each gender, and $6$ seats:

one beside the driver, two with their backs to the driver, and two behind them, facing the driver,

No two of the same gender were allowed to sit side by side.

The only people who were able to drive were the men.

So, how many ways could they all be seated?

The number of ways of seating the party is $144$.

There are three choices of drivers and three choices of women to sit beside him.

Anyone else can sit with their back directly to the driver; there are four choices remaining.

The one beside them must be of a different gender; there are two choices remaining.

The last two people have two ways to fill in the remaining seats.

Therefore the number of ways to seat the party is:

$3 \times 3 \times 4 \times 2 \times 2 = 144$

$\blacksquare$