Chapter 12.6, Problem 62PPE

To determine

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

Answer to Problem 62PPE

The statement is always true.

Explanation of Solution

Given information:

  $C{}_1=n$

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 ,

  $C{}_1=n$

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

The statement is always true.

Let $n=1,2,3,4,5,\mathrm{.......}$ .

Taking $n=10$ we get ,

Left Hand Side -

  $\begin{array}{l}C{}_{\left(10-1\right)}=\frac{10!}{\left(10-1\right)!\left(10-10+1\right)!}\\ =\frac{10!}{1!\left(10-1\right)!}\\ =\frac{10!}{9!}\\ =10\end{array}$

Right Hand Side -

  $n=10$

This implies L.H.S = R.H.S.

Hence , the statement is always true.