Let 10 Bags \(B_{1},B_{2},\dots,B_{10}\) which contain 21, 22, \dots, 30 different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is:
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The combination sum identity $\sum_{k=r}^n {}^{k}C_r = {}^{n+1}C_{r+1}$ is often called the Hockey-Stick Identity. Here, since our series starts at ${}^{21}C_{10}$ instead of the baseline ${}^{10}C_{10}$, the missing section can be cleanly accounted for by subtracting the trailing component ${}^{21}C_{11}$!
Step 1: Understanding the Concept:
To bring out 10 articles from a bag, we must select one bag and then select 10 articles from the articles it contains. The total ways is the sum of choices from each bag individually. Step 2: Key Formula or Approach:
1. Total ways \( S = \sum_{k=21}^{30} \binom{k}{10} \).
2. Use the identity \( \sum_{i=r}^n \binom{i}{r} = \binom{n+1}{r+1} \). Step 3: Detailed Explanation:
The sum is \( S = \binom{21}{10} + \binom{22}{10} + \dots + \binom{30}{10} \).
We can write this as:
\[ S = \left[ \binom{10}{10} + \binom{11}{10} + \dots + \binom{30}{10} \right] - \left[ \binom{10}{10} + \binom{11}{10} + \dots + \binom{20}{10} \right] \]
Applying the summation identity \( \sum_{k=10}^{n} \binom{k}{10} = \binom{n+1}{11} \):
\[ S = \binom{31}{11} - \binom{21}{11} \]
Using the property \( \binom{n}{r} = \binom{n}{n-r} \):
\[ S = \binom{31}{31-11} - \binom{21}{21-11} = \binom{31}{20} - \binom{21}{10} \]
Step 4: Final Answer:
The total ways is \( ^{31}C_{20} - ^{21}C_{10} \).