There are `5` different colour balls and `5` boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is

Options

A. `40`
B. `44`
C. `45`
D. `60`

Explanation

To solve the problem of placing 5 different colored balls into 5 boxes of the same colors such that no ball goes into the box of its own color, we can use the principle of derangements. A derangement is a permutation of elements such that none of the elements appear in their original position.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 balls (let's denote them as B1, B2, B3, B4, B5) and 5 boxes (let's denote them as Box1, Box2, Box3, Box4, Box5) where each ball has a corresponding box of the same color. We need to find the number of ways to place these balls in the boxes such that no ball is placed in its corresponding box.

2. **Derangement Formula**: The number of derangements (denoted as !n) of n items can be calculated using the formula:
\[ !n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]
For our case, n = 5.

3. **Calculate 5!**: First, we calculate the factorial of 5:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

4. **Calculate the Summation**: Now we calculate the summation part:
\[ \sum_{i=0}^{5} \frac{(-1)^i}{i!} = \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} + \frac{(-1)^5}{5!} \]
Evaluating each term:
- For i = 0: \(\frac{1}{1} = 1\)
- For i = 1: \(\frac{-1}{1} = -1\)
- For i = 2: \(\frac{1}{2} = 0.5\)
- For i = 3: \(\frac{-1}{6} \approx -0.1667\)
- For i = 4: \(\frac{1}{24} \approx 0.04167\)
- For i = 5: \(\frac{-1}{120} \approx -0.00833\)

Adding these values together:
\[ 1 - 1 + 0.5 - 0.1667 + 0.04167 - 0.00833 \approx 0.36667 \]

5. **Final Calculation of Derangement**: Now substituting back into the derangement formula:
\[ !5 = 120 \times 0.36667 \approx 44 \]

6. **Conclusion**: The number of ways to place the balls into the boxes such that no ball is placed in its own colored box is:
\[ \boxed{44} \]