Chapter 12, Problem 15PTTS

To determine

To find: To the possibility of how many arrangements using $7$ tiles

Answer to Problem 15PTTS

There are $5040$ arrangements are possible using 7 tiles.

Explanation of Solution

Given information: A player has $7$ tiles, each with a different letter

Calculation: First arrange these seven tiles without any constraints. So the total number of arrangements to do this is $7!$ , as we have total seven tiles.

Total number of arrangements $=7!$

  $\begin{array}{l}= 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =5040\end{array}$

So 5040 arrangements are possible

To determine

To find: The number of arrangements $5$ using all 7 tiles

Answer to Problem 15PTTS

There are 1260 arrangements 5 using all 7 tiles

Explanation of Solution

Given information: A player has 7 tiles, each with a different letter and using 5 in all 7 tiles

Calculation: According to the question there are 7 tiles and using 5 tiles

It means $(7-5)=2$ tiles are still constant

The number of arrangements using 5 in all 7 tiles is

  $\frac{7!}{2\times (7-5)!}$

  $\begin{array}{l}=\frac{7!}{2\times 2!}\\ =\frac{7!}{4}\\ =1260\end{array}$

So, there are 1260 arrangements 5 using all 7 tiles