Question

a) State the binomial theorem.

b) Explain how to prove the binomial theorem using a combinatorial argument.

c) Find the coefficient of${\mathit{x}}^{\mathbf{100}}{\mathit{y}}^{\mathbf{101}}$in the expansion of$\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{5}\mathbf{y}{\mathbf{)}}^{\mathbf{201}}$.

Short answer

(a) The binomial theorem is represented as${\left(\mathrm{x}+\mathrm{y}\right)}^{\mathrm{n}}=\sum _{\mathrm{j}=0}^{\mathrm{n}}\left(\begin{array}{c}\mathrm{n}\\ \mathrm{j}\end{array}\right)·{\mathrm{x}}^{\mathrm{n}-\mathrm{j}}]·{\mathrm{y}}^{\mathrm{j}}·$ .

(b) The quantity on the left represents the multiplication of$\left(\mathrm{x}+\mathrm{y}\right)$ with itself$\left(\mathrm{n}-1\right)$ times in a row, resulting in monomials with distinct powers of x and y multiplied to each other. In a given monomial, the exponents of two variables must always be of the type ${\mathrm{x}}^{\mathrm{n}-\mathrm{j}}\cdot {\mathrm{y}}^{\mathrm{j}}$.

(c) The coefficient of$x^{100}{y}^{101}$ in the expansion of${\left(2\mathrm{x}+5\mathrm{y}\right)}^{201}$ is $\left(\begin{array}{l}201\\ 101\end{array}\right)\cdot 2^{100}\cdot 5y^{101}$.

Step-by-step solution

Concept Introduction

The Binomial Theorem is a technique for expanding an equation raised to any finite power. A binomial Theorem is a useful expansion technique that can be used in Algebra, probability, and other fields.

Binomial Theorem

(a)

Let x and y be variables and let n be a nonnegative integer. Then –

$(x+y{)}^n=\sum _{j=0}^n \left(\begin{array}{c}n\\ j\end{array}\right)\cdot x^{n-j}\cdot y^j$

Therefore, the binomial theorem is obtained as$(x+y{)}^n=\sum _{j=0}^n \left(\begin{array}{c}n\\ j\end{array}\right)\cdot x^{n-j}\cdot y^j$ .

Binomial Theorem using Combinatorial Argument

(b)

The quantity on the left denotes the multiplication of$(\mathrm{x}+\mathrm{y})$ with itself successive$\left(\mathrm{n}-1\right)$ times, which gives rise to monomials where different powers of x and y are multiplied to each other. The exponents of two variables in a certain monomial must always be of the form${\mathrm{x}}^{\mathrm{n}-\mathrm{j}}\cdot {\mathrm{y}}^{\mathrm{j}}$ . Distributive property suggests that j of the n terms in the product must contribute a y to this monomial while others must contribute an x. Thus, the coefficient of this monomial is the number of ways to choose j elements from a set of size n, which is $\left(\begin{array}{c}\mathrm{n}\\ \mathrm{j}\end{array}\right)$, for all possible values of j.

Therefore, the result is obtained as${\mathrm{x}}^{\mathrm{n}-\mathrm{j}}.{\mathrm{y}}^{\mathrm{j}}$ .

Finding the coefficient

(c)

The${102}^{\mathrm{nd}}$ term in the expansion of$(2\mathrm{x}+5\mathrm{y}{)}^{201}\text{is}\left(\begin{array}{l}201\\ 101\end{array}\right)\cdot (2\mathrm{x}{)}^{100}(5\mathrm{y}{)}^{101}$ is , so the coefficient of${\mathrm{x}}^{100}{\mathrm{y}}^{101}$ is$\left(\begin{array}{l}201\\ 101\end{array}\right)\cdot 2^{100}\cdot 5{\mathrm{y}}^{101}$ .

Therefore, the result is obtained as $\left(\begin{array}{l}201\\ 101\end{array}\right)\cdot 2^{100}\cdot 5y^{101}$.