Counting Bit Strings with Specific Conditions

The problem asks us to find the number of bit strings of length ten that satisfy a specific condition: they must either start with a '1' bit or end with two '00' bits. A bit string is a sequence of 0s and 1s. The length of the string is ten.

To solve this, we can use the Principle of Inclusion-Exclusion for counting sets. Let S be the set of all bit strings of length ten. The total number of bit strings of length ten is $2^{10}$, since each of the ten positions can be either 0 or 1 independently. $2^{10} = 1024$.

Let A be the set of bit strings of length ten that start with a '1'.

Let B be the set of bit strings of length ten that end with two bits '00'.

We are looking for the number of strings that are in A or in B, which is given by $|A \cup B|$. According to the Principle of Inclusion-Exclusion:

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Let's calculate each term:

Calculate |A|: Strings Starting with 1

For a bit string of length ten to start with '1', the first bit is fixed as '1'. The remaining nine positions can be either 0 or 1. There are 9 remaining positions, and each has 2 possibilities (0 or 1). So, the number of such strings is $2^9$.

$$|A| = 2^9 = 512$$

Calculate |B|: Strings Ending with 00

For a bit string of length ten to end with '00', the last two bits are fixed as '0' and '0'. The first eight positions can be either 0 or 1. There are 8 remaining positions, and each has 2 possibilities. So, the number of such strings is $2^8$.

$$|B| = 2^8 = 256$$

Calculate |A $\cap$ B|: Strings Starting with 1 AND Ending with 00

For a bit string of length ten to start with '1' AND end with '00', the first bit is '1', and the last two bits are '00'. This fixes three positions (the first and the last two). The remaining $10 - 3 = 7$ positions can be either 0 or 1. There are 7 remaining positions, each with 2 possibilities. So, the number of such strings is $2^7$.

$$|A \cap B| = 2^7 = 128$$

Apply the Inclusion-Exclusion Principle

Now, we plug the values of $|A|$, $|B|$, and $|A \cap B|$ into the formula:

$$|A \cup B| = |A| + |B| - |A \cap B|$$

$$|A \cup B| = 512 + 256 - 128$$

$$|A \cup B| = 768 - 128$$

$$|A \cup B| = 640$$

Therefore, there are 640 bit strings of length ten that either start with a '1' bit or end with two bits '00'.

Step-by-Step Calculation Summary

Here is a summary of the steps taken:

1. Identify the two conditions: starts with 1 (Set A) and ends with 00 (Set B).
2. Understand the goal is to find the size of the union of these two sets, $|A \cup B|$.
3. Recall the Principle of Inclusion-Exclusion: $|A \cup B| = |A| + |B| - |A \cap B|$.
4. Calculate the number of strings starting with 1 ($|A|$). This fixes the first bit, leaving 9 free bits ($2^9$).
5. Calculate the number of strings ending with 00 ($|B|$). This fixes the last two bits, leaving 8 free bits ($2^8$).
6. Calculate the number of strings starting with 1 AND ending with 00 ($|A \cap B|$). This fixes the first and last two bits, leaving 7 free bits ($2^7$).
7. Substitute the calculated values into the Inclusion-Exclusion formula: $512 + 256 - 128$.
8. Perform the final calculation: $768 - 128 = 640$.

Revision Table: Counting Bit Strings

Additional Information: Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion is a counting technique used to find the number of elements in the union of two or more sets. For two sets, A and B, it states that the size of their union is the sum of their individual sizes minus the size of their intersection. This is because simply adding $|A|$ and $|B|$ would count the elements that are in both sets (the intersection) twice. Subtracting $|A \cap B|$ corrects this double-counting.

In the context of counting bit strings, we define sets based on properties (like starting with 1 or ending with 00) and then use this principle to find the number of strings that satisfy at least one of the properties.

Key concepts related to this problem include:

• Bit String: A sequence of 0s and 1s.
• Length of a String: The number of bits in the sequence.
• Combinatorics: The branch of mathematics dealing with combinations of objects belonging to finite sets in accordance with certain constraints.
• Counting Principles: Techniques like the multiplication rule (used to calculate $2^n$) and the Principle of Inclusion-Exclusion are fundamental in combinatorics.

This method can be extended to more than two sets as well. For three sets A, B, and C, the principle is: $|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |A \cap C| + |B \cap C|) + |A \cap B \cap C|$.