https://file.aiquickdraw.com/mathbot/file/1733719451951dasxw0ed.jpg

This problem involves a summation that appears to follow a systematic pattern involving combinations. Let’s carefully analyze and simplify the expression.

---

### The given expression:
\[
1 \cdot 2010 + 2 \cdot 2009 + 3 \cdot 2008 + \cdots + k \cdot (2011 - k) + \cdots + 2009 \cdot 2 + 2010 \cdot 1
\]

### General Term:
The \(k\)-th term is of the form:
\[
k \cdot (2011 - k)
\]
Expanding this:
\[
k \cdot 2011 - k^2
\]
So, the sum of the series becomes:
\[
\sum_{k=1}^{2010} k \cdot (2011 - k) = \sum_{k=1}^{2010} (k \cdot 2011 - k^2)
\]

This splits into two sums:
\[
2011 \sum_{k=1}^{2010} k - \sum_{k=1}^{2010} k^2
\]

---

### Evaluate Each Component:
1. **Sum of first \(n\) natural numbers**:
\[
\sum_{k=1}^n k = \frac{n(n+1)}{2}
\]
For \(n = 2010\):
\[
\sum_{k=1}^{2010} k = \frac{2010 \cdot 2011}{2}
\]

2. **Sum of squares of first \(n\) natural numbers**:
\[
\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}
\]
For \(n = 2010\):
\[
\sum_{k=1}^{2010} k^2 = \frac{2010 \cdot 2011 \cdot 4021}{6}
\]

---

### Substitute Back:
\[
\sum_{k=1}^{2010} k \cdot (2011 - k) = 2011 \cdot \frac{2010 \cdot 2011}{2} - \frac{2010 \cdot 2011 \cdot 4021}{6}
\]

Factor out common terms:
\[
= \frac{2010 \cdot 2011}{2} \left(2011 - \frac{2 \cdot 4021}{6}\right)
\]
Simplify the second term:
\[
2011 - \frac{4021}{3} = \frac{3 \cdot 2011 - 4021}{3} = \frac{2012}{3}
\]
So the expression becomes:
\[
\frac{2010 \cdot 2011}{2} \cdot \frac{2012}{3}
\]
\[
= \frac{2010 \cdot 2011 \cdot 2012}{6}
\]

---

### Recognizing Combinations:
\[
\frac{2010 \cdot 2011 \cdot 2012}{6} = \binom{2012}{3}
\]

---

### Final Answer:
The value of the expression is:
\[
\boxed{\binom{2012}{3}}
\]

Thus, the correct option is **(B)**.

---

Would you like me to provide further explanations or expand on any step? Here are five related questions:

1. What is the general formula for combinations and its applications in summations?
2. How can the sum of cubes of natural numbers be calculated?
3. Can this problem be visualized geometrically or combinatorially?
4. What are other scenarios where such patterns of summation arise?
5. How would this problem change if \(k\) started at a different number?

**Tip:** When dealing with summations, look for patterns or rewrite terms to exploit standard formulas for sums and squares.

The value of 1.2010 + 2.2009 + 3.2008 + ... + k.(2011-k) + ... + 2009.2 + 2010.1 equals to?

Explore the solution to the summation problem 1.2010 + 2.2009 + ... + k.(2011-k) with detailed steps. This involves arithmetic series, combinatorics, and binomial coefficients. Learn the derivation and understand the core concepts behind the solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation