How many bit-strings of length 9 will either start with ``1" or end with ``00"?

(A)64

(B)256

(C)320

(D)384

A. A

B. B

C. C

D. D

To solve the problem of finding how many bit-strings of length 9 either start with "1" or end with "00," let's break it down using the principle of inclusion and exclusion.

1. **Total number of bit-strings of length 9:**
   There are \( 2^9 = 512 \) total bit-strings of length 9, since each bit can be either 0 or 1.

2. **Bit-strings that start with "1":**
   If the bit-string starts with "1," the remaining 8 bits can be anything. So there are \( 2^8 = 256 \) bit-strings that start with "1."

3. **Bit-strings that end with "00":**
   If the bit-string ends with "00," the remaining 7 bits can be anything. So there are \( 2^7 = 128 \) bit-strings that end with "00."

4. **Bit-strings that start with "1" and end with "00":**
   If the bit-string starts with "1" and ends with "00," the remaining 6 bits can be anything. So there are \( 2^6 = 64 \) bit-strings that start with "1" and end with "00."

5. **Apply the principle of inclusion and exclusion:**
   - Bit-strings that start with "1" or end with "00" = (bit-strings that start with "1") + (bit-strings that end with "00") - (bit-strings that start with "1" and end with "00")
   - This gives us \( 256 + 128 - 64 = 320 \).

Thus, the correct answer is **C. 320**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How many bit-strings of length 10 start with "1" or end with "000"?
2. What is the probability that a random bit-string of length 9 starts with "1"?
3. How many bit-strings of length 9 do not start with "1" and do not end with "00"?
4. If a bit-string of length 9 is randomly generated, what is the probability that it ends with "00"?
5. How many bit-strings of length 8 contain exactly two "1"s?

**Tip:** The principle of inclusion and exclusion is useful for solving problems where you're asked to count elements that satisfy at least one of several conditions.

Learn how to count bit-strings of length 9 that either start with '1' or end with '00'. This problem uses combinatorial principles, including the principle of inclusion and exclusion, to find the correct answer. Suitable for advanced high school students.

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