For every integer n > 2 the sum of the expansions \(1 - {}^n{C_1} + {}^n{C_2} + - - - {( - 1)^n}{}.^n{C_n}\) is______
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Understanding the Sum of Binomial Expansions for n > 2
The question asks for the sum of the series \(1 - {}^n{C_1} + {}^n{C_2} - {}^n{C_3} + \dots + {( - 1)^n}{}.^n{C_n}\) for any integer \(n > 2\).
This specific sum looks very much like a standard result from the Binomial Theorem. Let's recall the Binomial Theorem.
Applying the Binomial Theorem
The Binomial Theorem states that for any real numbers \(a\) and \(b\), and any non-negative integer \(n\):
\[ (a+b)^n = \sum_{k=0}^{n} {{}^n{C_k} a^{n-k} b^k} \]
Expanding this, we get:
\[ (a+b)^n = {}^n{C_0} a^n b^0 + {}^n{C_1} a^{n-1} b^1 + {}^n{C_2} a^{n-2} b^2 + \dots + {}^n{C_n} a^0 b^n \]
Now, let's consider the expansion of \((1-x)^n\). We can write \((1-x)^n\) as \((1+(-x))^n\). Using the Binomial Theorem with \(a=1\) and \(b=-x\):
\[ (1+(-x))^n = \sum_{k=0}^{n} {{}^n{C_k} (1)^{n-k} (-x)^k} \]
\[ (1-x)^n = {}^n{C_0} (1)^n (-x)^0 + {}^n{C_1} (1)^{n-1} (-x)^1 + {}^n{C_2} (1)^{n-2} (-x)^2 + \dots + {}^n{C_n} (1)^0 (-x)^n \]
Simplifying the terms:
\[ (1-x)^n = {}^n{C_0} (1) - {}^n{C_1} (x) + {}^n{C_2} (x^2) - {}^n{C_3} (x^3) + \dots + {( - 1)^n}{}.^n{C_n} (x^n) \]
\[ (1-x)^n = {}^n{C_0} - {}^n{C_1} x + {}^n{C_2} x^2 - {}^n{C_3} x^3 + \dots + {( - 1)^n}{}.^n{C_n} x^n \]
Evaluating the Sum
The given sum is \(1 - {}^n{C_1} + {}^n{C_2} - {}^n{C_3} + \dots + {( - 1)^n}{}.^n{C_n}\). We know that \({}^n{C_0} = 1\). So the sum can be written as:
\[ {}^n{C_0} - {}^n{C_1} + {}^n{C_2} - {}^n{C_3} + \dots + {( - 1)^n}{}.^n{C_n} \]
Comparing this with the expansion of \((1-x)^n\), we can see that this is exactly the expansion of \((1-x)^n\) evaluated at \(x=1\).
Let's substitute \(x=1\) into the expansion of \((1-x)^n\):
\[ (1-1)^n = {}^n{C_0} - {}^n{C_1} (1) + {}^n{C_2} (1)^2 - {}^n{C_3} (1)^3 + \dots + {( - 1)^n}{}.^n{C_n} (1)^n \]
\[ (0)^n = {}^n{C_0} - {}^n{C_1} + {}^n{C_2} - {}^n{C_3} + \dots + {( - 1)^n}{}.^n{C_n} \]
So, the sum is equal to \((0)^n\). The question states that \(n > 2\). For any integer \(n > 2\), \(0^n = 0\).
Thus, the sum of the expansion is \(0\).
Checking the Options
Let's look at the given options:
| Option | Value | Matches Result? |
|---|---|---|
| 1 | \(n(n - 1)2^{n - 2}\) | No |
| 2 | \(0\) | Yes |
| 3 | \(n 2^{m - 1}\) | No (contains undefined variable 'm') |
| 4 | \((^{2n}{C_n})^2\) | No |
The calculated sum is \(0\), which matches Option 2.
Conclusion
For every integer \(n > 2\), the sum of the expansion \(1 - {}^n{C_1} + {}^n{C_2} - {}^n{C_3} + \dots + {( - 1)^n}{}.^n{C_n}\) is \(0\).
Revision Table: Key Concepts
| Concept | Description | Formula/Example |
|---|---|---|
| Binomial Coefficient | \({}^n{C_k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements. | \({}^n{C_k} = \frac{n!}{k!(n-k)!}\) |
| Binomial Theorem | Expands powers of a binomial expression \((a+b)^n\). | \((a+b)^n = \sum_{k=0}^{n} {{}^n{C_k} a^{n-k} b^k}\) |
| Sum of Alternating Binomial Coefficients | The sum \(\sum_{k=0}^{n} {(-1)^k {}^n{C_k}}\) is the expansion of \((1-1)^n\). | \({}^n{C_0} - {}^n{C_1} + {}^n{C_2} - \dots + (-1)^n {}^n{C_n} = (1-1)^n\) |
Additional Information: Properties of Binomial Coefficients
Binomial coefficients have many interesting properties. Here are a few related to sums:
- Sum of all binomial coefficients: \(\sum_{k=0}^{n} {{}^n{C_k}} = {}^n{C_0} + {}^n{C_1} + \dots + {}^n{C_n} = (1+1)^n = 2^n\). This is the expansion of \((1+x)^n\) at \(x=1\).
- Sum of even-indexed coefficients: \({}^n{C_0} + {}^n{C_2} + {}^n{C_4} + \dots = 2^{n-1}\) for \(n \ge 1\).
- Sum of odd-indexed coefficients: \({}^n{C_1} + {}^n{C_3} + {}^n{C_5} + \dots = 2^{n-1}\) for \(n \ge 1\).
- Vandermonde's Identity: \(\sum_{k=0}^{r} {{}^m{C_k} {}^n{C_{r-k}}} = {}^{m+n}{C_r}\). This identity is useful in combinatorial proofs.
The alternating sum property used in the solution, \(\sum_{k=0}^{n} {(-1)^k {}^n{C_k}} = 0\) for \(n \ge 1\), is a direct application of setting \(x=-1\) in the binomial expansion of \((1+x)^n\).
Important Questions from Binomial Expansion
If $x = \frac{1}{4}$, then the greatest term in the expansion of $(2 + 3x)^{15}$ will be
- What is the number of distinct terms in the expansion of $(p + q + r + s)^n$, where $n \in \mathbb{N}$?
Consider the expansion of (1 + x) n. Let p, q, r and s be the coefficients of first, second, nth and (n + 1)th terms respectively. What is (ps + qr) equal to?
What is the sum of the coefficients of first and last terms in the expansion of (1 + x) 2n , where n is a natural number?
- What is \(\displaystyle\sum_{r=0}^n\) 2 r C(n, r) equal to ?
