If \(\frac{{^{11}C_1}}{2} + \frac{{^{11}C_2}}{3} + \ldots + \frac{{^{11}C_9}}{10} = \frac{n}{m}\) with \(\gcd(n, m) = 1\), then \(n + m\) is equal to:
If \(\frac{{^{11}C_1}}{2} + \frac{{^{11}C_2}}{3} + \ldots + \frac{{^{11}C_9}}{10} = \frac{n}{m}\) with \(\gcd(n, m) = 1\), then \(n + m\) is equal to:
Solution and Explanation
Step 1: Rewrite the Sum in Terms of a Series
The sum can be expressed as:
\[ \sum_{r=1}^{9} \left( ^{11}C_r \right) \cdot \left( ^{11}C_{r+1} \right) \]
Step 2: Simplify the Series Using Combinatorial Identities
This can be simplified by recognizing a pattern and using properties of binomial coefficients:
\[ = \frac{1}{12} \sum_{r=1}^{9} \left( ^{12}C_{r+1} \right) \]
Further simplifying, we get:
\[ = \frac{1}{12} \left[2^{12} - 26\right] = \frac{2035}{6} \]
Step 3: Determine \(n\) and \(m\)
From the simplified result, \(n = 2035\) and \(m = 6\), with \(\gcd(n, m) = 1\).
Step 4: Calculate \(n + m\)
\[ n + m = 2035 + 6 = 2041 \]
So, the correct answer is: 2041
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.