In how many ways can 20 articles be packed in the parcels so that the first contains 8 articles, the second 7 and the third 5?

Answer

To solve the problem of packing 20 articles into three parcels containing 8, 7, and 5 articles respectively, we can use the concept of permutations and combinations. Here’s the step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Total Number of Articles**:
We have a total of 20 articles.

2. **Determine the Distribution of Articles**:
The articles need to be packed into three parcels:
- First parcel: 8 articles
- Second parcel: 7 articles
- Third parcel: 5 articles

3. **Calculate the Total Arrangements**:
The total number of ways to arrange 20 articles is given by \(20!\) (20 factorial).

4. **Account for Identical Groups**:
Since the articles in each parcel are indistinguishable within the same parcel, we need to divide by the factorial of the number of articles in each parcel to avoid counting identical arrangements multiple times. Thus, we divide by \(8!\) for the first parcel, \(7!\) for the second, and \(5!\) for the third.

5. **Write the Formula**:
The total number of ways to pack the articles can be expressed as:
\[ \text{Total Ways} = \frac{20!}{8! \times 7! \times 5!} \]

6. **Calculate the Factorials**:
- Calculate \(20!\)
- Calculate \(8!\)
- Calculate \(7!\)
- Calculate \(5!\)

7. **Substitute the Values**:
Substitute the calculated factorials into the formula to find the total number of ways.

8. **Final Calculation**:
Perform the final calculation to arrive at the answer.

### Final Answer:
The total number of ways to pack the articles is:
\[ \frac{20!}{8! \times 7! \times 5!} \]