7 white balls and 3 black balls are kept randomly in order. Find the probability that no two adjacent balls are black.

Options

A. `1//2`
B. `7//15`
C. `2//15`
D. `1//3`

Answer

To solve the problem of finding the probability that no two adjacent balls are black when we have 7 white balls and 3 black balls, we can follow these steps:

### Step 1: Calculate the Total Arrangements of Balls
We have a total of 10 balls (7 white and 3 black). The total number of arrangements of these balls can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{10!}{7! \times 3!} \]

### Step 2: Calculate the Favorable Arrangements
To ensure that no two black balls are adjacent, we can first arrange the 7 white balls. The arrangement of the white balls creates 8 possible slots for the black balls (one before each white ball, one after the last white ball):

- W W W W W W W
- _ _ _ _ _ _ _ _ (8 slots)

We need to choose 3 out of these 8 slots to place the black balls. The number of ways to choose 3 slots from 8 is given by:

\[ \text{Ways to choose slots} = \binom{8}{3} \]

### Step 3: Calculate the Total Favorable Outcomes
Since the black balls are indistinguishable, we do not need to multiply by any additional factorial for the black balls. Thus, the total number of favorable arrangements is simply:

\[ \text{Favorable arrangements} = \binom{8}{3} \]

### Step 4: Calculate the Probability
The probability that no two adjacent balls are black is given by the ratio of the number of favorable arrangements to the total arrangements:

\[ P(\text{no two adjacent black balls}) = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{\binom{8}{3}}{\frac{10!}{7! \times 3!}} \]

### Step 5: Simplify the Expression
Now we can simplify this expression. First, calculate \(\binom{8}{3}\):

\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

Next, calculate the total arrangements:

\[ \frac{10!}{7! \times 3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

Now, substitute these values back into the probability formula:

\[ P(\text{no two adjacent black balls}) = \frac{56}{120} = \frac{7}{15} \]

### Final Answer
Thus, the probability that no two adjacent balls are black is:

\[ \frac{7}{15} \]

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