Key Concept: Sum of Products of Binomial Coefficients using Binomial Expansions

Many sums involving products of binomial coefficients can be elegantly solved by considering the coefficient of a specific power of ${x}$ in the product of two binomial expansions. This technique is a direct application of the Binomial Theorem and is closely related to Vandermonde's Identity.

Problem: Find the value of ${S = \sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}} }$

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Detailed Solution:

1. Transform the Sum using the Symmetry Property of Binomial Coefficients

The given sum is: ${S = {}^{22}{C_0}{}^{23}{C_0} + {}^{22}{C_1}{}^{23}{C_1} + \dots + {}^{22}{C_{22}}{}^{23}{C_{22}}}$

We use the property of binomial coefficients which states that ${{}^n{C_r} = {}^n{C_{n - r}}}$ (the number of ways to choose ${r}$ items from ${n}$ is the same as choosing ${n-r}$ items to leave behind).

Applying this property to the second term, ${{}^{23}{C_r}}$, we replace it with ${{}^{23}{C_{23 - r}}}$: ${S = \sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_{23 - r}}} }$

Why this step? This transformation is crucial because it ensures that the sum of the lower indices of the binomial coefficients (${r}$ and ${23-r}$) is a constant value (${r + (23-r) = 23}$). This constant sum indicates that our total sum will be the coefficient of ${x^{23}}$ in a suitable product of binomial expansions.

Expanding the modified sum: ${S = {}^{22}{C_0}{}^{23}{C_{23}} + {}^{22}{C_1}{}^{23}{C_{22}} + {}^{22}{C_2}{}^{23}{C_{21}} + \dots + {}^{22}{C_{22}}{}^{23}{C_1}}$

2. Relate the Sum to the Coefficient in a Binomial Product

Consider the binomial expansions of ${(1+x)^{22}}$ and ${(1+x)^{23}}$:

From the Binomial Theorem, ${(1+x)^n = \sum_{k=0}^{n} {^n C_k} x^k}$.

For ${(1+x)^{22}}$: ${(1+x)^{22} = {}^{22}{C_0} + {}^{22}{C_1}x + {}^{22}{C_2}x^2 + \dots + {}^{22}{C_r}x^r + \dots + {}^{22}{C_{22}}x^{22}}$

For ${(1+x)^{23}}$: ${(1+x)^{23} = {}^{23}{C_0} + {}^{23}{C_1}x + {}^{23}{C_2}x^2 + \dots + {}^{23}{C_k}x^k + \dots + {}^{23}{C_{23}}x^{23}}$

Now, let's consider the product of these two expansions: ${(1+x)^{22}(1+x)^{23} = \left( \sum_{r=0}^{22} {}^{22}{C_r} x^r \right) \left( \sum_{k=0}^{23} {}^{23}{C_k} x^k \right)}$

We are looking for the coefficient of ${x^{23}}$ in this product. A term ${x^{23}}$ in the product is formed by multiplying a term ${({}^{22}{C_r} x^r)}$ from the first expansion with a term ${({}^{23}{C_k} x^k)}$ from the second expansion such that ${r+k=23}$. This means ${k = 23-r}$.

So, the coefficient of ${x^{23}}$ in the product is the sum of terms of the form ${{}^{22}{C_r}{}^{23}{C_{23-r}}}$ for all possible values of ${r}$. Since ${0 \le r \le 22}$ (from ${{}^{22}{C_r}}$), and ${0 \le 23-r \le 23}$ (from ${{}^{23}{C_{23-r}}}$), the valid range for ${r}$ is ${0 \le r \le 22}$.

Therefore, the coefficient of ${x^{23}}$ in ${(1+x)^{22}(1+x)^{23}}$ is precisely our sum $${S = \sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_{23 - r}}} }$.

3. Simplify the Product and Find the Coefficient

Using the rules of exponents, we can simplify the product: ${(1+x)^{22}(1+x)^{23} = (1+x)^{22+23} = (1+x)^{45}}$

Now, we need to find the coefficient of ${x^{23}}$ in the expansion of ${(1+x)^{45}}$. According to the Binomial Theorem, the coefficient of ${x^{k}}$ in ${(1+x)^n}$ is ${{}^n{C_k}}$. Here, ${n=45}$ and ${k=23}$.

Thus, the coefficient of ${x^{23}}$ in ${(1+x)^{45}}$ is ${{}^{45}{C_{23}}}$

Since our sum ${S}$ is equal to this coefficient, we have: ${S = {}^{45}{C_{23}}}$

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Tips and Common Mistakes:

• Vandermonde's Identity: This problem is a direct application of Vandermonde's Identity, which states: ${ \sum_{k=0}^{R} {m \choose k} {n \choose R-k} = {m+n \choose R} }$ In our case, with ${m=22}$, ${n=23}$, ${R=23}$, and ${k=r}$, the sum matches this identity exactly: ${ \sum_{r=0}^{22} {}^{22}{C_r} {}^{23}{C_{23-r}} = {}^{22+23}{C_{23}} = {}^{45}{C_{23}} }$ Recognizing this identity can save time.
• Symmetry Property: Always remember the symmetry property ${{}^n{C_r} = {}^n{C_{n - r}}}$ as it's fundamental for transforming sums into the form required for coefficient extraction or Vandermonde's Identity.
• Checking Limits: Be careful with the limits of the summation. Here, ${r}$ goes from 0 to 22. This matches the maximum index for ${{}^{22}{C_r}}$ and ensures that ${23-r}$ stays within the valid range for ${{}^{23}{C_{23-r}}}$ (${23-0=23}$ and ${23-22=1}$).
• Incorrect Power of x: A common mistake is to choose the wrong power of ${x}$ to find the coefficient of. The sum of the lower indices of the binomial coefficients after applying the symmetry property (${r + (23-r) = 23}$) tells you the power of ${x}$ to look for.

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Summary:

To evaluate the sum ${ \sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}} }$:

1. We first transform the sum using the symmetry property of binomial coefficients (${{}^n{C_r} = {}^n{C_{n - r}}}$) to get { \sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_{23 - r}}} }$. This step ensures that the sum of the lower indices is constant ({23}$$).
2. We then recognize this sum as the coefficient of ${x^{23}}$ in the product of the binomial expansions of ${(1+x)^{22}}$ and ${(1+x)^{23}}$.
3. The product ${(1+x)^{22}(1+x)^{23}}$ simplifies to ${(1+x)^{45}}$.
4. Finally, using the Binomial Theorem, the coefficient of ${x^{23}}$ in ${(1+x)^{45}}$ is ${{}^{45}{C_{23}}}$

The value of the sum is ${{}^{45}{C_{23}}}$.