Number of ways in which 5 different toys can be distributed in 5 children if exactly one child does not get any toy is greater than

Options

A. 1200
B. 2400
C. 960
D. 300

Explanation

To solve the problem of distributing 5 different toys among 5 children such that exactly one child does not receive any toy, we can follow these steps:

### Step 1: Choose the child who will not receive a toy
We have 5 children, and we need to select 1 child who will not receive any toy. The number of ways to choose 1 child from 5 is given by the combination formula:

\[ \text{Number of ways to choose 1 child} = \binom{5}{1} = 5 \]

**Hint:** Think of this step as simply selecting one child from the group of five.

### Step 2: Distribute the toys among the remaining children
Now that we have chosen one child who will not receive a toy, we are left with 4 children. We need to distribute 5 different toys among these 4 children. Since each child must receive at least one toy, we can use the principle of inclusion-exclusion or the "stars and bars" method.

To ensure that each of the 4 children gets at least 1 toy, we can first give 1 toy to each of the 4 children. This uses up 4 toys, leaving us with 1 toy to distribute freely among the 4 children.

### Step 3: Distributing the remaining toy
The remaining toy can be given to any of the 4 children. Therefore, there are 4 choices for distributing this last toy.

**Hint:** After ensuring each child has at least one toy, consider how many options you have for the remaining toys.

### Step 4: Calculate the total arrangements
The total number of ways to distribute the toys can be calculated by multiplying the number of ways to choose the child who does not receive a toy by the number of ways to distribute the toys among the remaining children.

The total number of ways is:

\[ \text{Total ways} = \binom{5}{1} \times 4^1 = 5 \times 4 = 20 \]

### Step 5: Arranging the toys
Since the toys are different, we also need to consider the arrangements of the toys. The number of ways to arrange 5 different toys is given by \(5!\):

\[ 5! = 120 \]

### Step 6: Final calculation
Now, we multiply the number of ways to choose the child and the arrangements of the toys:

\[ \text{Total arrangements} = 20 \times 120 = 2400 \]

Thus, the total number of ways to distribute 5 different toys among 5 children such that exactly one child does not receive any toy is **2400**.

**Final Answer:** 2400