Out of 15 balls, of which some are white and the rest are black, how many should be white so that the number of ways in which the balls can be arranged in a row may be the greatest possible? It is assumed that the balls of same color are alike?

Options

A. `6 or 9`
B. `7 or 8`
C. `4 or 11`
D. `5 or 10`

Answer

To find the number of white balls that maximizes the arrangements of 15 balls (some white and some black), we can follow these steps:

### Step 1: Define Variables
Let \( r \) be the number of white balls. Therefore, the number of black balls will be \( 15 - r \).

### Step 2: Write the Formula for Arrangements
The total number of arrangements (permutations) of these balls can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{15!}{r! \cdot (15 - r)!} \]

### Step 3: Identify the Maximum Arrangements
To maximize the number of arrangements, we need to find the value of \( r \) that maximizes the expression \( \frac{15!}{r! \cdot (15 - r)!} \). This expression is equivalent to \( 15C_r \), which represents the number of ways to choose \( r \) white balls from 15.

### Step 4: Use Combinatorial Properties
The value of \( 15C_r \) is maximized when \( r \) is as close as possible to half of 15. Since 15 is odd, the two closest integers are 7 and 8.

### Step 5: Conclusion
Thus, the number of white balls that maximizes the arrangements is either 7 or 8.

### Final Answer
The number of white balls should be either 7 or 8 to maximize the arrangements of the balls.

---