Number of ways in which 5 plus (+) signs and 5 minus (-) signs be arranged in a row so that no two minus signs are together is

Options

A. 6
B. 7
C. 8
D. 10

Answer

To solve the problem of arranging 5 plus (+) signs and 5 minus (-) signs in a row such that no two minus signs are together, we can follow these steps:

### Step-by-Step Solution:

1. **Arrange the Plus Signs**:
First, we arrange the 5 plus signs in a row. This can be visualized as follows:
```
+ + + + +
```

2. **Identify the Gaps**:
After placing the plus signs, we can identify the gaps where the minus signs can be placed. The arrangement of the plus signs creates 6 potential gaps for the minus signs:
```
_ + _ + _ + _ + _ + _
```
Here, the underscores (_) represent the gaps where we can place the minus signs.

3. **Choose Gaps for Minus Signs**:
We need to place 5 minus signs in these 6 gaps, ensuring that no two minus signs are together. Since we have 6 gaps and we need to choose 5 of them, we can calculate the number of ways to choose 5 gaps from 6. This can be expressed using the combination formula:
\[ \text{Number of ways} = \binom{6}{5} \]

4. **Calculate the Combination**:
The combination \(\binom{6}{5}\) can be calculated as:
\[ \binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5! \cdot 1!} = \frac{6 \times 5!}{5! \times 1} = 6 \]

5. **Conclusion**:
Therefore, the total number of ways to arrange 5 plus signs and 5 minus signs such that no two minus signs are together is:
\[ \text{Total arrangements} = 6 \]

### Final Answer:
The number of ways to arrange 5 plus signs and 5 minus signs such that no two minus signs are together is **6**.

---