In how many ways 5 plus (+) signs and 5 minus (-) signs be arranged in a row so that no two minus signs are together?

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. Since all the plus signs are identical, there is only 1 way to arrange them.

**Arrangement**: + + + + +

2. **Identify the Gaps**:
Next, we need to identify the gaps where we can place the minus signs. When we arrange the 5 plus signs, we create gaps around and between them.

The gaps are:
- Before the first plus sign
- Between the first and second plus signs
- Between the second and third plus signs
- Between the third and fourth plus signs
- Between the fourth and fifth plus signs
- After the fifth plus sign

This gives us a total of 6 gaps.

3. **Select Gaps for Minus Signs**:
We need to place the 5 minus signs in these 6 gaps, ensuring that no two minus signs are in the same gap (to prevent them from being together).

We can choose 5 gaps out of the 6 available gaps to place the minus signs.

4. **Calculate Combinations**:
The number of ways to choose 5 gaps from 6 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of gaps and \( r \) is the number of gaps we want to choose.

Therefore, we need to calculate \( \binom{6}{5} \).

\[ \binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5! \cdot 1!} = \frac{6 \times 5!}{5! \times 1} = 6 \]

5. **Conclusion**:
Thus, the total number of ways to arrange 5 plus signs and 5 minus signs in a row such that no two minus signs are together is **6**.

### Final Answer:
The total number of arrangements is **6**.

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