In how many ways can 6 pictures be hung from 4 picture nails on a wall?

Answer

To solve the problem of how many ways 6 pictures can be hung from 4 picture nails on a wall, we can break it down into a few clear steps.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 6 pictures and 4 nails. We need to choose 4 pictures from the 6 available and then arrange those 4 pictures on the 4 nails.

2. **Choosing Pictures**: First, we need to select 4 pictures from the 6 available. 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. In this case, we will calculate \( \binom{6}{4} \).

\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} \]

3. **Calculating \( \binom{6}{4} \)**: We can simplify this calculation:
\[ \binom{6}{4} = \frac{6 \times 5}{2 \times 1} = \frac{30}{2} = 15 \]

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

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

5. **Total Arrangements**: To find the total number of ways to hang the pictures, we multiply the number of ways to choose the pictures by the number of ways to arrange them:

\[ \text{Total Ways} = \binom{6}{4} \times 4! = 15 \times 24 = 360 \]

### Final Answer:
Thus, the total number of ways to hang 6 pictures from 4 picture nails on a wall is **360**.

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