Question:
Suppose $\displaystyle\sum_{r=0}^{2023} r^{22023} C_r=2023 \times \alpha \times 2^{2022}$ Then the value of $\alpha$ is
Suppose $\displaystyle\sum_{r=0}^{2023} r^{22023} C_r=2023 \times \alpha \times 2^{2022}$ Then the value of $\alpha$ is
Updated On: Mar 31, 2026
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Correct Answer: 1012
Approach Solution - 1
The correct answer is 1012.
using result
Then
using result
Then
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Approach Solution -2
Using the result:
\[
\sum_{r=0}^{n} \binom{n}{r} \cdot C_r = n \cdot (n+1) \cdot 2^{n-2},
\]
for \( n = 2023 \), we have:
\[
\sum_{r=0}^{2023} \binom{2023}{r} \cdot C_r = 2023 \cdot 2024 \cdot 2^{2021}.
\]
Equating:
\[
2023 \cdot \alpha \cdot 2^{2022} = 2023 \cdot 2024 \cdot 2^{2021}.
\]
Simplify:
\[
\alpha = \frac{2024}{2} = 1012.
\]
Final Answer: 1012.
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Concepts Used:
Combinations
The method of forming subsets by selecting data from a larger set in a way that the selection order does not matter is called the combination.
- It means the combination of about ‘n’ things taken ‘k’ at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.
- For example, Imagine you go to a restaurant and order some soup.
- Five toppings can complement the soup, namely:
- croutons,
- orange zest,
- grated cheese,
- chopped herbs,
- fried noodles.
But you are only allowed to pick three.
- There can be several ways in which you can enhance your soup with savory.
- The selection of three toppings (subset) from the five toppings (larger set) is called a combination.
Use of Combinations:
It is used for a group of data (where the order of data doesn’t matter).