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# The value of $C_1^2 + C_2^2 + ... + C_n^2$ (where ${C_i}$ is the ${i^{th}}$ coefficient of ${(1 + x)^n}$ expansion) is:

Last updated date: 25th Jul 2024
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Hint:The given equation is based on a special case of the binomial theorem. We can simply expand the equation using the binomial theorem, then square on both the sides and arrive at the answer using the formula: ${(1 + x)^n} = \sum\limits_{}^n {_{r - 0}n{C_r}{x^r}} = {C_0} + {C_1} + {C_2}{x^2} + ...{C_n}{x_n}$ . Finally, we need to find the value of $C_1^2 + C_2^2 + ... + C_n^2$ by the above formula.

The Binomial Theorem is a technique for extending an expression elevated to some finite power. A binomial Theorem is a useful expansion method that can be used in Algebra, probability, and other fields. A binomial expression is an algebraic expression which contains two dissimilar terms. Example- $a + b,{a^3} + {b^3}$ etc.

Binomial Theorem can be explained as-
If $n \in N,x,y \in R$ then ${(x + y)^n}{ = ^n}{\sum _{r = 0}}n{C_r}{x^{n - r}}{y^r}$
where $n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$
Points to be noted are:
-The total number of terms in the expansion of ${(x + y)^n}$ are $(n + 1)$.
-The sum of exponents of $x$ and $y$ is always $n$.
-$n{C_0},n{C_1},n{C_2},...,n{C_n}$ are called binomial coefficients and also represented by ${C_0},{C_1},{C_2},...,{C_n}$.
-The binomial coefficients which are equidistant from the beginning and from the ending are equal i.e. $n{C_0} = n{C_n},n{C_1} = n{C_{n - 1}},n{C_2} = n{C_{n - 2}}$ etc.

We can expand the equation with the help of formula as follows:
${(1 + x)^n} = \sum\limits_{}^n {_{r = 0}n{C_r}{x^r}} = {C_0} + {C_1} + {C_2}{x^2} + ...{C_n}{x_n}$
$\Rightarrow {(1 + x)^n} = n{C_0} + x(n{C_1}) + {x^2}(n{C_2}) + ... + {x_n}(n{C_n})$
Now if we square the equation, on the left-hand side we will get ${(1 + x)^{2n}}$.
The coefficient of ${x^n}$ in the equation ${(1 + x)^{2n}}{ = ^{2n}}{C_n}$
Hence, we will get the squared equation as follows:
$^{2n}{C_n} = {(n{C_0})^2} + {(n{C_1})^2} + {(n{C_2})^2} + ... + {(n{C_n})^2}$
Where ${C_i}$ is the ${i^{th}}$ coefficient of ${(1 + x)^n}$ expansion.

Therefore, we can conclude that: $C_1^2 + C_2^2 + ... + C_n^2{ = ^{2n}}{C_n} = \dfrac{{2n!}}{{n!n!}}$.

Note: Here we have assumed $n$ to be a rational number and $x$ be a real number such that $\left| x \right| < 1$. To find binomial coefficients we can also use Pascal’s Triangle. Binomial coefficients refer to the integers which are coefficients in the binomial theorem.
$C_1^2 + C_2^2 + ... + C_n^2{ = ^{2n}}{C_n} = \dfrac{{2n!}}{{n!n!}}$ is one of the most important properties of binomial coefficient.