Finite Math problem...

[one_voice]

How many 6 digit Delaware license plates are there with three distinct digits, one occurring once, the second occurring twice and the third occurring thrice? The digit 0 cannot be the first digit.

This is one of my son's math problems...he's having trouble..not sure whether to use combinations or permutations? Or if these choices are right.

Any ideas? It reads like a foreign language to me.

[petronius]

1. 9 x 8 x 7 - nope, not right...

It's really the same problem as the 'how many 3 digit combinations are possible' - theyve just gussied it up a bit. But there's just 3 possible slots, and 9 possibilities for the first slot...

Edit - sorry, I misread the 0 thing, though it said zero couldn't be used. Do they mean 0 can't be the first number on the plate, or zero can't be the number that occurs once? I'll have to think a bit more...

[one_voice]

3. 0 cannot be the first digit on the plate...

It can be used for any "occurrence" just not the first digit. Thanks for the help. I was able to help with multiplication, division, algebra, but I have no idea about this.

[dimbear]

2. Hints:

The first thing to do is count how many patterns there are. The digit which appears once could be in 6 places. The digit which appears twice has to choose 2 places from the remaining 5, which is the combination 5 choose 2, which is 10. The digit which appears three times is stuck, there are no more choices, have to fill the remaining slots.

So there are 60 patterns.

You could choose the single digit 10 ways, the twice appearing one 9 ways, and the thrice appearing 8 ways. So there are 720 ways to choose the digits. Total so far: 60 x 720.

But wait: what about the restriction that 0 can't start? Well, there's just as much chance of starting with zero as any other number. So 10% are thrown out by the restriction.

You end up with 9/10 times 60 times 720.

[one_voice]

4. Thanks...

that's helpful.

[struggle4progress]

5. Just go through it slowly. Here's a complete solution, but your son will do better

to suffer through at least some of the discovery himself

There are six slots to occupy, but only three distinct numbers to put in the slots, and 0 cannot go into the first slot

How can you describe such a license plate?
You can say first whether the first digit on the plate occurs once, twice, or thrice
If the first digit occurs once, you can say what it is. Then there is a digit that appears twice: you can say where it occurs and what it is. Finally you say what digit occurs in the remaining three slots
If the first digit occurs twice, you can say what it is and where else it occurs. Then there is a digit that appears once: you can say where it occurs and what it is. Finally you say what digit occurs in the remaining three slots
If the first digit occurs thrice, you can say what it is and where else it occurs. Then there is a digit that appears once: you can say where it occurs and what it is. Finally you say what digit occurs in the remaining two slots

So split the problem into three simpler problems (with all the above restrictions):
How many license plates are there with the number in the first slot occurring once?
How many license plates are there with the number in the first slot occurring twice?
How many license plates are there with the number in the first slot occurring thrice?
The sum of the answers to these simpler problems will be the answer to the original problem

How many license plates are there with the number in the first slot occurring once?
0 cannot occur in the first slot, so there are 9 possibilities for the number in first slot
We now need to pick 2 more slots for the number that occurs twice: there are 5-choose-2 = 10 ways to choose this slot
We now need to choose a digit to occupy these slots: it must be different from the number in the first slot, so there are 9 possibilities for the number we choose to put in these 2 slots
We now need to choose a digit to occupy the remaining 3 slots: it must be different from the two numbers we have already chosen, so there are 8 possibilities for the number we choose to put in these 3 slots
Total: 9x10x9x8 = 9*720 = 6480

How many license plates are there with the number in the first slot occurring twice?
0 cannot occur in the first slot, so there are 9 possibilities for the number in first slot
We need to pick another slot to drop this number into: there are 5 ways to choose it
We now need to pick another (unused) slot for the number that occurs once: there are 4 ways to choose it
We now need to choose a digit to occupy the remaining 3 slots: it must be different from the two numbers we have already chosen, so there are 8 possibilities for the number we choose to put in these 3 slots
Total: 9x5x4x8 = 9*180 = 1620
<correction: 9x5x4x8 = 9*160 = 1440>

How many license plates are there with the number in the first slot occurring thrice?
0 cannot occur in the first slot, so there are 9 possibilities for the number in first slot
We need to pick two more slots to drop this number into: there are 5-choose-2 = 10 ways to choose them
We now need to pick another (unused) slot for the number that occurs once: there are 3 ways to choose it
We now need to choose a digit to occupy the remaining 2 slots: it must be different from the two numbers we have already chosen, so there are 8 possibilities for the number we choose to put in these 2 slots
Total: 9x10x3x2 = 9*60 = 540

Answer: 6480 + 1620 + 540
<correction: 6480 + 1440 + 540>

[one_voice]

6. My son says thanks a lot...

this really helps. You guys are the best.

[some guy]

7. this part is wrong

Total: 9x5x4x8 = 9*180 = 1620

9 x 5 = 45
45 x 4 = 180
180 x 8 = 1440

[struggle4progress]

8. Thanks! "Always check a mathematician's arithmetic, because it's probably wrong"