Chapter 12.6, Problem 63PPE

To determine

To show: whether statement is always , sometimes, or true. Assume $n\ne 0$ .

Answer to Problem 63PPE

The statement is sometimes true.

Explanation of Solution

Given information:

  $C{}_x>x$

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 ,

  $C{}_x>x$

  $C{}_x=\frac{3!}{x!\left(3-x\right)!}$

The value of x be $x=1,2,3$ because at other values the factorial gets negative.

Let $x=2$

Left Hand Side −

  $\begin{array}{l}C{}_2=\frac{3!}{2!\left(3-2\right)!}\\ =\frac{3!}{2!1!}\\ =\frac{3\times 2\times 1}{2\times 1\times 1}\\ =3\end{array}$

Right Hand Side −

  $x=2$

Therefore , L.H.S > R.H.S.

Hence, the statement is sometimes true.