In how many ways 7 pictures can be hanged on 9 pegs?

Answer

To solve the problem of how many ways 7 pictures can be hanged on 9 pegs, we can follow these steps:

### Step 1: Understand the Problem
We have 7 pictures and 9 pegs. We need to determine how many different ways we can hang these pictures on the pegs.

### Step 2: Choose the Pegs
First, we need to select which 7 pegs out of the 9 will be used to hang the pictures. The number of ways to choose 7 pegs from 9 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

In this case, we calculate:
\[ \binom{9}{7} \]

### Step 3: Calculate the Combination
Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
we can substitute \( n = 9 \) and \( r = 7 \):
\[ \binom{9}{7} = \frac{9!}{7!(9-7)!} = \frac{9!}{7! \cdot 2!} \]

### Step 4: Simplify the Combination
Now we can simplify this expression:
\[ \binom{9}{7} = \frac{9 \times 8}{2 \times 1} = \frac{72}{2} = 36 \]

### Step 5: Arrange the Pictures
After selecting the 7 pegs, we need to arrange the 7 pictures on these pegs. The number of ways to arrange 7 pictures is given by \( 7! \) (7 factorial).

Calculating \( 7! \):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

### Step 6: Calculate the Total Ways
Now, we multiply the number of ways to choose the pegs by the number of ways to arrange the pictures:
\[ \text{Total Ways} = \binom{9}{7} \times 7! = 36 \times 5040 \]

### Step 7: Final Calculation
Calculating the final result:
\[ 36 \times 5040 = 181440 \]

### Conclusion
Therefore, the total number of ways to hang 7 pictures on 9 pegs is **181440**.

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