Chapter 8.4, Problem 112E

To determine

To evaluate: The term $C{}_3{\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^7$

Answer to Problem 112E

  $0.2503$

Explanation of Solution

Given information:

The expansion of ${\left(\frac{1}{4}+\frac{3}{4}\right)}^7$ .

Concept Used:

Pascal’s triangle:

  

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

First evaluate $C{}_3$

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, $C{}_3$ is the 3^rd element of the 10^th row of the Pascal’s triangle.

The binomial coefficients from the tenth row of Pascal’s Triangle are 1, 10, 45, 120, 210, 252, 210, 120, 45, 10,1.

The third element of this row is 120.

So, $C{}_3=120$

Now consider,

  $\begin{array}{c}C{}_3{\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^7=\left(120\right){\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^7\\ =120\left(\frac{1}{64}\right)\left(\frac{2187}{16,384}\right)\\ \approx 0.2503\end{array}$