Six '+' signs and five '-' signs are to be arranged in a row. If the arrangement is at random, then the probability that no 2 '-' sings are together, is

Options

A. `1/32`
B. `3/32`
C. `21/22`
D. `1/22`

Answer

To solve the problem of arranging six '+' signs and five '-' signs in a row such that no two '-' signs are together, we can follow these steps:

### Step 1: Arrange the '+' signs
First, we arrange the six '+' signs. The arrangement of these signs creates gaps where we can place the '-' signs.

When we arrange 6 '+' signs, we can visualize it as:
```
+ + + + + +
```
This arrangement creates 7 gaps (including the ends) where the '-' signs can be placed:
```
_ + _ + _ + _ + _ + _ + _
```
Here, the underscores represent the gaps.

### Step 2: Place the '-' signs in the gaps
To ensure that no two '-' signs are together, we need to select 5 out of the 7 available gaps to place the '-' signs.

The number of ways to choose 5 gaps from 7 is given by the combination formula:
\[ \text{Number of ways} = \binom{7}{5} \]
This can also be calculated as:
\[ \binom{7}{5} = \binom{7}{2} = \frac{7!}{5! \cdot 2!} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 3: Calculate the total arrangements
The total arrangements of the '+' and '-' signs without any restrictions is given by the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{(6+5)!}{6! \cdot 5!} = \frac{11!}{6! \cdot 5!} = 462 \]

### Step 4: Calculate the probability
Now we can find the probability that no two '-' signs are together:
\[ \text{Probability} = \frac{\text{Number of favorable arrangements}}{\text{Total arrangements}} = \frac{21}{462} \]

To simplify this fraction:
\[ \frac{21}{462} = \frac{1}{22} \]

### Final Answer
Thus, the probability that no two '-' signs are together is:
\[ \frac{1}{22} \]