A block is a consecutive sequence of the same digit that is not part of a longer such sequence.
a.) Determine the number of sequences of 0's and 1's of length 10 in which every block of 1's has an even length.
b.) Determine the number of sequences of 0's and 1's of length 10 in which every block of 1's has an odd length.
To determine the number of sequences of 0's and 1's of length 10 that satisfy the given conditions, we can use the concept of counting techniques and break down the problem into smaller steps.
a.) To find the number of sequences in which every block of 1's has an even length:
Step 1: Determine the number of blocks of 1's.
In this case, since every block of 1's has an even length, we need to consider the number of even blocks of 1's. Since each block can have a length of 2, 4, 6, or 8, we have four options for the length of each block.
Step 2: Determine the number of 0's.
Now that we know the number of blocks of 1's, we need to count the number of 0's. Each 0 will create a separate block on its own.
Step 3: Combine the values from Step 1 and Step 2.
Multiply the number of different block lengths (4 options) by the number of blocks in total. Then, multiply this value by 2 (the number of choices for each block, 0 or 1) raised to the power of the number of 0's.
Example calculation:
Let's say we have 3 blocks of 1's in our sequence, with block lengths of 4, 2, and 6. Also, we have 4 0's in the sequence.
Step 1: Number of blocks of 1's = 3
Step 2: Number of 0's = 4
Step 3: Total number of sequences = 4 (block lengths) * 3 (number of blocks) * 2^4 (number of 0's) = 4 * 3 * 16 = 192
Therefore, the total number of sequences of 0's and 1's of length 10, where every block of 1's has an even length, is 192.
b.) To find the number of sequences in which every block of 1's has an odd length:
The approach for part b is similar to part a, except that in this case, we need to consider blocks of 1's with odd lengths.
Step 1: Determine the number of blocks of 1's.
In this case, since every block of 1's has an odd length, we need to consider the number of odd blocks of 1's. Since each block can have a length of 1, 3, 5, 7, or 9, we have five options for the length of each block.
Step 2: Determine the number of 0's.
Count the number of 0's in the sequence. Each 0 will create a separate block on its own.
Step 3: Combine the values from Step 1 and Step 2.
Multiply the number of different block lengths (5 options) by the number of blocks in total. Then, multiply this value by 2 raised to the power of the number of 0's.
Example calculation:
Let's say we have 2 blocks of 1's in our sequence, with block lengths of 3 and 9. Additionally, we have 6 0's in the sequence.
Step 1: Number of blocks of 1's = 2
Step 2: Number of 0's = 6
Step 3: Total number of sequences = 5 (block lengths) * 2 (number of blocks) * 2^6 (number of 0's) = 5 * 2 * 64 = 640
Therefore, the total number of sequences of 0's and 1's of length 10, where every block of 1's has an odd length, is 640.