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:
- Understanding the problem:
We have to arrange six - signs and four - signs in a line such that no two - signs occur together.
- Strategy to solve the problem:
To solve this problem, we can use the concept of permutations. We can first arrange the six - signs in a line, leaving gaps for the four - signs. Then, we can place the four - signs in the gaps such that no two - signs are together.
- Calculating the total number of ways:
1. Arrange the six - signs: There are 7 positions where the four - signs can be placed (5 between the - signs and 2 at the ends).
This can be done in 7! ways.
2. Place the four - signs: We have to choose 4 positions out of the 7 available positions to place the four - signs. This can be done in 7C4 ways.
3. Multiply the number of ways: To find the total number of ways, we multiply the number of ways to arrange the - signs with the number of ways to place the - signs.
Total number of ways = 7! * 7C4
- Calculating the final answer:
Now, we can calculate the total number of ways using the formula mentioned above.
- Final Answer:
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 7! * 7C4 = 7! * (7*6*5) / (4*3*2) = 5040 * 35 = 176400 ways.