Chapter 8.4, Problem 116E

To determine

To evaluate: The term $C{}_4{\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^4$ .

Answer to Problem 116E

  $C{}_4{\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^4=0.273$

Explanation of Solution

Given information:

A binomial is given as ${\left(\frac{1}{2}+\frac{1}{2}\right)}^8$ .

Concept Used:

Pascal’s triangle:

  

The n^th row of Pascal’s triangle gives the coefficients of ${\left(x+y\right)}^n$ .

First evaluate $C{}_4$

Now, in the Pascal’s triangle $C{}_r$ represents the r^th element of the n^th row of the Pascal’s triangle for $r=0,1,2,\mathrm{...},n$ .

So, $C{}_4$ is the 4^th element of the 8^th row of the Pascal’s triangle.

The binomial coefficients from the eighth row of Pascal’s Triangle are 1, 8, 28, 56, 70, 56, 28, 8,1.

The fourth element of this row is 70.

So, $C{}_4=70$

Now consider,

  $\begin{array}{c}C{}_4{\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^4=\left(70\right){\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^4\\ =70\left(\frac{1}{16}\right)\left(\frac{1}{16}\right)\\ \approx 0.273\end{array}$