The number of ways in which 4 pictures can be hung from 6, picture nails on the wall is :

Options

A. `4^6`
B. `""^(4) P_6`
C. `""^(6) P_4`
D. `6^4`s

Answer

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

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to hang 4 pictures using 6 available picture nails. The order in which the pictures are hung matters since they are distinct.

2. **Choosing the Nails**:
First, we need to choose 4 nails from the 6 available nails. The number of ways to choose 4 nails from 6 can be calculated using the combination formula \( nCr \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
\[ \text{Number of ways to choose 4 nails from 6} = \binom{6}{4} \]

3. **Calculating the Combinations**:
The combination formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Applying this to our case:
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

4. **Arranging the Pictures**:
After choosing the 4 nails, we can arrange the 4 pictures on these nails. The number of ways to arrange 4 distinct pictures is given by \( 4! \) (4 factorial).
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

5. **Calculating the Total Arrangements**:
The total number of ways to hang the 4 pictures on the chosen nails is the product of the number of ways to choose the nails and the number of arrangements of the pictures:
\[ \text{Total ways} = \binom{6}{4} \times 4! = 15 \times 24 = 360 \]

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