The number of ways in which 5 picturers can be hung from 7 picture nails on the wall is

Options

A. `7^(5)`
B. `5^(7)`
C. 2520
D. none of these

Answer

To solve the problem of how many ways 5 pictures can be hung from 7 picture nails on the wall, we can break it down into a few steps.

### Step-by-Step Solution:

1. **Understand the Problem**: We have 5 pictures and 7 picture nails. We need to select 5 nails from the 7 available nails to hang the pictures.

2. **Select the Nails**: The first step is to choose 5 nails from the 7. This can be done using 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. Here, we need to calculate \( \binom{7}{5} \).

\[ \binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} \]

3. **Calculate \( \binom{7}{5} \)**:

\[ \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

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

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

5. **Total Ways**: The total number of ways to hang the pictures is the product of the number of ways to choose the nails and the number of ways to arrange the pictures.

\[ \text{Total Ways} = \binom{7}{5} \times 5! = 21 \times 120 = 2520 \]

### Final Answer:
The total number of ways in which 5 pictures can be hung from 7 picture nails on the wall is **2520**.

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