Chapter 5.7, Problem 30PPSE

Solution

To explain : The reason that the coefficient of the given expression do not match the numbers in the third row of Pascal’s Triangle.

The coefficient of the given expression do not match the numbers in the third row of Pascal’s Triangle due to the difference in the coefficients of ${\left(a+b\right)}^n$ and ${\left(x+2y\right)}^3$ .

Given information :

The binomial is: ${\left(x+2y\right)}^3$ .

Formula used:

Binomial Theorem: Pascal's Triangle is used to expand binomial powers. For a positive integer $n,$

  ${\left(a+b\right)}^n=P_0{a}^n+P_1{a}^{n-1}b+P_2{a}^{n-2}{b}^2+\mathrm{......}+P_{n-1}a{b}^{n-1}+P_n{b}^n$ .

Where $P_0,P_1,\mathrm{.....},P_n$ are the coefficients of the $nth$ row of the Pascal's Triangle.

Explanation :

In ${\left(a+b\right)}^n$ , the coefficient of $a\text{ and }b$ is $1$ whereas in ${\left(x+2y\right)}^3$ , the coefficient of $y$ is $2$ . So, while expanding the given binomial expression, the elements in the third row of the Pascal's Triangle will be multiplied by $2^0,2^1,2^2,\mathrm{......}$ .

Construct the Pascale’s Triangle up to third row as below:

  $\begin{array}{l}1\\ 11\\ 121\\ 1331\end{array}$

So, the expansion of ${\left(a+b\right)}^3$ will be:

  $\begin{array}{c}{\left(a+b\right)}^3=1{\left(a\right)}^3+3{\left(a\right)}^2\left(b\right)+3\left(a\right){\left(b\right)}^2+1{\left(b\right)}^3\\ =a^3+3a^{2}b+3a{b}^2+b^3\end{array}$

But the expansion of ${\left(x+2y\right)}^3$ will be:

  $\begin{array}{c}{\left(x+2y\right)}^3=1{\left(x\right)}^3+3{\left(x\right)}^2\left(2y\right)+3\left(x\right){\left(2y\right)}^2+1{\left(2y\right)}^3\\ =x^3+6x^{2}y+12x{y}^2+8y^3\\ =1\left(2^0\right)x^3+3\left(2^1\right)x^{2}y+3\left(2^2\right)x{y}^2+1\left(2^3\right)y^3\end{array}$

Therefore, the coefficient of the given expression do not match the numbers in the third row of Pascal’s Triangle due to the difference in the coefficients of ${\left(a+b\right)}^n$ and ${\left(x+2y\right)}^3$ .