There are 10 bags B1,B2,B3,...,B10,which contain 21, 22? 30 different articles respectively. The total number of ways to bring out 10 articles from a bag is

Answer

To solve the problem of determining the total number of ways to bring out 10 articles from 10 bags (B1, B2, ..., B10) containing 21, 22, ..., 30 different articles respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of articles in each bag:**
- Bag B1 contains 21 articles.
- Bag B2 contains 22 articles.
- Bag B3 contains 23 articles.
- Bag B4 contains 24 articles.
- Bag B5 contains 25 articles.
- Bag B6 contains 26 articles.
- Bag B7 contains 27 articles.
- Bag B8 contains 28 articles.
- Bag B9 contains 29 articles.
- Bag B10 contains 30 articles.

2. **Determine the combinations for selecting articles from each bag:**
- The number of ways to choose 10 articles from each bag can be represented using the binomial coefficient (nCr), where n is the number of articles in the bag and r is the number of articles we want to choose.
- Therefore, the total ways to choose 10 articles from each bag can be expressed as:
- From B1: \( \binom{21}{10} \)
- From B2: \( \binom{22}{10} \)
- From B3: \( \binom{23}{10} \)
- From B4: \( \binom{24}{10} \)
- From B5: \( \binom{25}{10} \)
- From B6: \( \binom{26}{10} \)
- From B7: \( \binom{27}{10} \)
- From B8: \( \binom{28}{10} \)
- From B9: \( \binom{29}{10} \)
- From B10: \( \binom{30}{10} \)

3. **Sum the combinations:**
- The total number of ways to bring out 10 articles from any of the bags is the sum of the combinations from each bag:
\[ \text{Total Ways} = \binom{21}{10} + \binom{22}{10} + \binom{23}{10} + \binom{24}{10} + \binom{25}{10} + \binom{26}{10} + \binom{27}{10} + \binom{28}{10} + \binom{29}{10} + \binom{30}{10} \]

4. **Use the identity for binomial coefficients:**
- We can use the identity that relates binomial coefficients:
\[ \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} \]
- This allows us to express the sums in a different form, but for our purpose, we can directly calculate the sum as stated.

5. **Calculate the individual terms:**
- Calculate each binomial coefficient using a calculator or software that can handle combinatorial calculations.

6. **Final Calculation:**
- Add all the calculated values together to get the final total number of ways to select 10 articles from the bags.