Suppose n different games are to be given to n children. In how many ways can this be done so that eactly one child gets no game.

Answer

To solve the problem of distributing n different games to n children such that exactly one child receives no game, we can follow these steps:

### Step 1: Choose the child who will receive no game
We have n children, and we need to choose one child who will not receive any game. The number of ways to choose one child from n children is given by:

\[ \text{Ways to choose the child} = n \]

### Step 2: Distribute the games among the remaining children
After selecting the child who will not receive a game, we have \(n - 1\) children left. We need to distribute all n games such that each of the \(n - 1\) children gets exactly one game, and one of these children will receive an extra game.

### Step 3: Choose the extra game
We can choose any one of the n games to be the extra game. The number of ways to choose one game from n games is:

\[ \text{Ways to choose the extra game} = n \]

### Step 4: Distribute the remaining games
After assigning the extra game to one of the \(n - 1\) children, we have \(n - 1\) games left to distribute among the \(n - 1\) children. Each child will receive exactly one game. The number of ways to distribute \(n - 1\) games to \(n - 1\) children is given by the factorial of the number of games (or children), which is:

\[ \text{Ways to distribute remaining games} = (n - 1)! \]

### Step 5: Combine all the choices
Now, we can combine all the choices made in the previous steps. The total number of ways to distribute the games such that exactly one child gets no game is given by:

\[ \text{Total ways} = (\text{Ways to choose the child}) \times (\text{Ways to choose the extra game}) \times (\text{Ways to distribute remaining games}) \]

Substituting the values we found:

\[ \text{Total ways} = n \times n \times (n - 1)! \]

This simplifies to:

\[ \text{Total ways} = n^2 \times (n - 1)! \]

### Final Result
Thus, the total number of ways to distribute n different games to n children such that exactly one child gets no game is:

\[ n^2 \times (n - 1)! \]

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