Problem Statement: Find the total number of ways in which six ' ' and four '-' signs can be arranged in a line such that no two '-' signs occur together.

Solution:

To solve this problem, we need to use the concept of permutation and combination. We can arrange six ' ' and four '-' signs in a line in the following way:

_ _ _ _ _ _ _ _ _ _ (total ten positions)

Now, we need to place the four '-' signs in such a way that no two '-' signs occur together. We can do this by selecting four positions out of the ten positions and placing '-' signs in those positions. The remaining positions will be filled with ' ' signs.

Step 1: Select four positions out of ten positions for '-' signs.

We can select four positions out of ten positions in 10C4 ways.

10C4 = (10 * 9 * 8 * 7)/(4 * 3 * 2 * 1) = 210

Step 2: Place '-' signs in the selected positions and ' ' signs in the remaining positions.

We can place '-' signs in the selected positions in 4! ways and ' ' signs in the remaining positions in 6! ways.

4! * 6! = 2,177,280

Step 3: Find the total number of ways in which six ' ' and four '-' signs can be arranged in a line such that no two '-' signs occur together.

The total number of ways in which six ' ' and four '-' signs can be arranged in a line such that no two '-' signs occur together is the product of the number of ways in which we can select four positions out of ten positions and the number of ways in which we can arrange the '-' and ' ' signs in those positions.

Total number of ways = 210 * 2,177,280 = 457,228,800

Therefore, the total number of ways in which six ' ' and four '-' signs can be arranged in a line such that no two '-' signs occur together is 457,228,800.