Chapter 12.6, Problem 59PPE

To determine

To calculate: the value of expression.

Answer to Problem 59PPE

The value is $4$ .

Explanation of Solution

Given information:

  $C{}_2+C{}_1+C{}_0$

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

Consider ,

  $C{}_2+C{}_1+C{}_0$

  $\begin{array}{l}C{}_2=\frac{2!}{2!\left(2-2\right)!}\\ =\frac{2!}{2!0!}\\ =\frac{2\times 1}{2\times 1\times 1}\\ =1\end{array}$

Solving second term,

  $\begin{array}{l}C{}_1=\frac{2!}{1!\left(2-1\right)!}\\ =\frac{2!}{1!1!}\\ =\frac{2\times 1}{1\times 1}\\ =2\end{array}$

Solving third term,

  $\begin{array}{l}C{}_0=\frac{2!}{0!\left(2-0\right)!}\\ =\frac{2!}{0!2!}\\ =\frac{2\times 1}{2\times 1\times 1}\\ =1\end{array}$

Now,

  $\begin{array}{l}C{}_2+C{}_1+C{}_0=1+2+1\\ =4\end{array}$

Hence, the value is $4.$