There are four different coloured balls and four boxes of the same colour as that of each ball is. Find the number of ways in which exactly one ball can be put in a box so that the colour of the box and that of ball is distinct.

Options

A. A) 16
B. B) 10
C. C) `"^4C_2`
D. D) none of these

Answer

To solve the problem of finding the number of ways to place exactly one ball in a box such that the color of the box and the ball are distinct, we can follow these steps:

### Step 1: Understand the Problem
We have 4 different colored balls and 4 boxes, each box matching the color of one of the balls. The goal is to place exactly one ball in a box such that the ball's color does not match the box's color.

### Step 2: Choose the Ball
First, we need to select which ball we will place in a box. Since there are 4 different colored balls, we have 4 choices.

### Step 3: Choose a Box
Next, we need to choose a box for the selected ball. Since the box must be of a different color than the ball, we can only choose from the remaining 3 boxes.

### Step 4: Calculate the Total Ways
Now, we can calculate the total number of ways to achieve this:
- For each of the 4 balls, we have 3 choices of boxes.
- Therefore, the total number of ways = Number of balls × Number of boxes available for each ball = 4 × 3 = 12.

### Final Answer
Thus, the number of ways in which exactly one ball can be put in a box so that the color of the box and that of the ball is distinct is **12**.

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