Q 8: Count the number of 01-strings with following constraints.

(c) The length is 9. Number of 1s is 3 more than number of 0s.

I hope you realize that 3 of the four questions that you posted have been answered before you made these reposts.

To count the number of 01-strings with the given constraints, follow these steps:

1. Start by assuming the number of 0s in the string. Since the number of 1s is 3 more than the number of 0s, let's assume there are x 0s in the string.

2. If there are x 0s in the string, then the number of 1s will be x + 3.

3. Since the length of the string is 9, we need to determine the positions of the 0s and 1s in the string. We can do this by using a combination of x 0s and (x + 3) 1s in a 9-character long string.

4. The number of ways to select x positions for the 0s from a total of 9 positions can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of positions and r is the number of positions to be chosen. In this case, n = 9 and r = x.

5. For each combination of positions for the 0s, there will be a corresponding allocation of positions for the 1s. The number of ways to arrange the remaining (x + 3) 1s in the remaining (9 - x) positions can be calculated using the permutation formula: P(n, r) = n! / (n-r)!, where n is the total number of positions and r is the number of positions to be allocated. In this case, n = 9 - x and r = (x + 3).

6. Multiply the number of combinations of positions for the 0s (from step 4) with the number of permutations of positions for the 1s (from step 5) to get the total number of 01-strings with x 0s and (x + 3) 1s.

7. Repeat steps 4 to 6 for different values of x, starting from 0 and going up to the largest possible value of x (when x + 3 = 9).

8. Add up the total number of 01-strings obtained from each iteration of steps 4 to 6 to get the final count of 01-strings with the given constraints.

Note: This process can be automated using programming languages like Python by implementing loops and mathematical functions like factorial and combination.