A child has 6 pockets. In how many ways can he put 5 marbles in his pocket?

Answer

To solve the problem of how many ways a child can put 5 marbles into 6 pockets, we can follow these steps:

### Step 1: Understand the Problem
The child has 5 indistinguishable marbles and 6 distinguishable pockets. We need to find the number of ways to distribute these marbles into the pockets.

### Step 2: Identify the Method
Since the marbles are indistinguishable, we can use the "stars and bars" combinatorial method. This method helps us find the number of ways to put n indistinguishable objects (marbles) into k distinguishable boxes (pockets).

### Step 3: Set Up the Equation
In the stars and bars method, we represent the marbles as stars and the separations between different pockets as bars. For 5 marbles and 6 pockets, we need to place 5 stars and 5 bars (since we need 5 dividers to create 6 sections).

### Step 4: Calculate the Total Arrangements
The total number of symbols (stars + bars) is:
- Stars (marbles) = 5
- Bars (dividers) = 5
- Total symbols = 5 + 5 = 10

Now, we need to choose positions for either the stars or the bars. We can choose positions for the 5 stars out of the 10 total positions:
\[ \text{Number of ways} = \binom{10}{5} \]

### Step 5: Calculate the Binomial Coefficient
Using the formula for binomial coefficients:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
we can calculate:
\[ \binom{10}{5} = \frac{10!}{5! \cdot 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

### Conclusion
Thus, the total number of ways the child can put 5 marbles into 6 pockets is **252**.

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