Combination Problem: 8 0s & 10 1s in Bit Strings

[MinusTheBear]

Homework Statement

I don't see why this is a combination problem, because to me dealing with bit strings means that order matters.
How many bit strings contain exactly eight 0s and 10 1s if every 0 must be immediately followed by a 1?

Homework Equations

The Attempt at a Solution

Since every 1 immediately has a 0 following it, I thought about each 1 containing a 0 right after it as one item. That is, "10" is a single entity. This means that I have eight "10"'s and two 0's. Making a total of 10 items.

Since it is a bit string, I thought order matters since 10 10 10 10 10 10 10 10 0 0 is different from 10 0 10 0 10 10 10 10 10 10. Therefore, it's a permutation.

So I have P(10,8) * P(2,2). This was wrong. The solution is C(10,8) * C(2,2).

I guess my question is, what does it mean that order doesn't matter? I must be misunderstanding what that means.

 

[Stephen Tashi]

I don't see why this is a combination problem, because to me dealing with bit strings means that order matters.
How many bit strings contain exactly eight 0s and 10 1s if every 0 must be immediately followed by a 1?

Order does matter in a string, but notice that a symbol such as "1" can be used in different places in a string and if you imagine two "1"'s swapping places in the string, the string isn't changed. So, in a manner of speaking, the order of the various "1"'s doesn't matter. If fact, you can't tell the "1"s apart.

Combinatorial problems often involve both formula for permutations and formula for combinations. Unfortunately, nobody knows a systematic algorithm for taking the English language version of a combinatorial problem and translating it into a mathematical expression for the answer. Combinatorial problems are like chess problems. There is (as yet) no simple algorithm for solving them. With practice you can recognize some groups of problems as being of the same type.

Since every 1 immediately has a 0 following it,

That's backwards of how you stated the problem.

I thought about each 1 containing a 0 right after it as one item.

That's a good idea. Think of writing the string as filling in a sequence of 10 boxes. You must pick 8 of the boxes to have a "10" in them (if we treat "10" as a single item). It doesn't matter in what order you choose to fill the 8 boxes, just so you get them all filled.

 

[MinusTheBear]

Order does matter in a string, but notice that a symbol such as "1" can be used in different places in a string and if you imagine two "1"'s swapping places in the string, the string isn't changed. So, in a manner of speaking, the order of the various "1"'s doesn't matter. If fact, you can't tell the "1"s apart.

Combinatorial problems often involve both formula for permutations and formula for combinations. Unfortunately, nobody knows a systematic algorithm for taking the English language version of a combinatorial problem and translating it into a mathematical expression for the answer. Combinatorial problems are like chess problems. There is (as yet) no simple algorithm for solving them. With practice you can recognize some groups of problems as being of the same type.
That's backwards of how you stated the problem.That's a good idea. Think of writing the string as filling in a sequence of 10 boxes. You must pick 8 of the boxes to have a "10" in them (if we treat "10" as a single item). It doesn't matter in what order you choose to fill the 8 boxes, just so you get them all filled.

Thanks, I see the error in my thinking, now. I think I just spent to much time doing homework and not enough time sleeping last night to think straight. I'm sort of in a mental fog. I appreciate it. Coincidentally, I had no issue with this in Probability and Stats, but its giving me issues in Discrete Math 2 even though the examples are simpler.