Solved: Counting Problems Homework - Distinct Digit Numbers, Even/Odd

[LostSeeker]

Homework Statement

----1. From the digits 0, 1, 2, 3, 4, 5, 6, how many four-digit numbers with distinct digits can be constructed? How many of these are even numbers?

----2. How many integers between 1000 and 9999 inclusive have distinct digits? Of these how many are odd numbers?

Homework Equations

Use the product rule, etc.

The Attempt at a Solution

----1. The first question is straight forward.

There are four-digit numbers: _ _ _ _
From left-to-right on possibilities: 6 (can't be zero; hence, 6 versus 7) * 6 * 5 * 4
Answer: 720 possibilities.

The second type of question is where the troubles begin.

Does anyone know of a simple online tutorial that would help? I'm sure this is all obvious but it is not to me.

The following is an attempt. . .

There are four-digit numbers: _ _ _ _
From right-to-left on possibilities: 3 (even numbers), 6 (even or odd), 5 (even or odd), 3 (even or odd but no zero)
Wrong answer: 270 possibilities.

Where am I going wrong?

----2. Again, the first part is easy for the second problem. The answer is 9*9*8*7 = 4,536.

The second part confuses me.
From right-to-left on possibilities: 5 (odd numbers), 9, 8, 6
Wrong answer: 5*9*8*6 = 2,160.

According to the textbook, from right-to-left the answer really is: 5, 8, 8, 7.
Why is this correct? Why those particular numbers?

 

[Sethric]

There are four-digit numbers: _ _ _ _
From right-to-left on possibilities: 3 (even numbers), 6 (even or odd), 5 (even or odd), 3 (even or odd but no zero)
Wrong answer: 270 possibilities.

Where am I going wrong?

0 is an even number, so you have to choose from 4 digits to start with. Then the rest changes depending on whether or not you chose 0. It will affect the thousands digit.

The second part confuses me.
From right-to-left on possibilities: 5 (odd numbers), 9, 8, 6
Wrong answer: 5*9*8*6 = 2,160.

According to the textbook, from right-to-left the answer really is: 5, 8, 8, 7.
Why is this correct? Why those particular numbers?

Are you sure it is right-to-left? I always place my digits depending where stipulations are. So I would place Ones, Thousands, (the order doesn't matter for the others). If your choice affects another position, I try to do that position first, unless there are shared dependencies.

 

[LostSeeker]

Thank you.

OK, I never thought of zero as an even number. When it comes to evens and odds, I've always considered them natural numbers. The ordering is as you say, as well.

----1. From the digits 0, 1, 2, 3, 4, 5, 6, how many four-digit numbers with distinct digits can be constructed? How many of these are even numbers?

Let me try this once more. . .

But this time I won't work right-to-left.

There are four-digit numbers: _ _ _ _
Ones position: 4 (even numbers)
Thousands position: 5 (non-zero, minus above selection)
Hundreds position: 5 (minus the two above selections)
Tens position: 4 (minus the three above selections)

That's not right, either. 420 is supposedly the answer.

----2. How many integers between 1000 and 9999 inclusive have distinct digits? Of these how many are odd numbers?

There are four-digit numbers: _ _ _ _
Ones position: 5 (odd numbers)
Thousands position: 8 (non-zero, minus above selection)
Hundreds position: 8 (minus the two above selections)
Tens position: 7 (minus the three above selections)

Answer: 2,240. This appears correct. If that is the case, why not the other problem?

 

[Sethric]

There are four-digit numbers: _ _ _ _
Ones position: 4 (even numbers)
Thousands position: 5 (non-zero, minus above selection)
Hundreds position: 5 (minus the two above selections)
Tens position: 4 (minus the three above selections)

I think you missed what I was saying by a little. It will affect the thousands digit if the ones digit was non-zero; it won't affect the thousands digit if it was zero. You need to do both cases and add them together.

 

[LostSeeker]

I see. It would be nice if my textbook actually explained the solutions to these problems. Instead it leaves the student in the dark.

Once again, I thank you.

There are four-digit numbers: _ _ _ _

Ones position: 1 (zero as even number)
Thousands position: 6 (anything but zero)
Hundreds position: 5 (minus above two numbers)
Tens position: 4 (minus above three numbers)

Ones position: 3 (non-zero even number)
Thousands position: 5 (non-zero number and minus above number)
Hundreds position: 5 (minus above two numbers)
Tens position: 4 (minus above three numbers)

Answer: 6*5*4*1 + 5*5*4*3 = 420.