# Perms and Combs HSC question (1 Viewer)

#### Rayser323

##### New Member
Can someone please help me with this question? I think its a past HSC question.

A motorway pay station has five toll gates, three of which are automatically operated and two of which are manually operated. Drivers with the exact money can use any one of the five gates, but drivers requiring change have to use one of the two manually operated gates. A Ford driver, a Holden driver and a Toyota driver use the motorway every day.
ii. On another day, all three drivers have the exact money. Find the number of ways in which the three drivers can go through the pay station so that exactly one goes through a manually operated gate.

#### ultra908

##### Active Member
Sorry in advance i dont know latex

First we pick who will go through the manual gate, i.e 3C1=3. We then pick which of the two manual gates they will go through, i.e. 3*2
Then, we arrange the 2 others in the 3 different gates, i.e. 3!
Thus the final answer is 3*2*3!=36

#### Rayser323

##### New Member
That logic makes sense but the answer is 54 :/
Im guessing the calculation is 3 x 2 x3 x3 but idk how they came about those numbers

#### kevin3314

##### New Member
2C1*(3C1)^(2)*3 = 54
When Toyota goes through the gate, they have 2 choices[Either manual gate](2C1), the other cars have 3 choices[Either automatic gates](3C1) and since two cars, it must be squared i.e. (3C1)^2
Since for all three cases, it is the same, just multiply by 3

#### fan96

##### 617 pages
Choose the one car that gets the manually operated gate, then choose which gate.

$\bg_white \binom{3}{1}\cdot \binom 21 = 6.$

The remaining two cars must go through the auto gates (of which there are three).

$\bg_white \binom 31 \cdot \binom 31 = 9.$

Finally, $\bg_white 6 \cdot 9 = 54$.

Then, we arrange the 2 others in the 3 different gates, i.e. 3!
A gate can be used by more than one car, so there's no need to do any arranging.

#### ultra908

##### Active Member
A gate can be used by more than one car, so there's no need to do any arranging.
ah yes stupid assumption on my part

Thank you!!