# Five books of different heights are arranged in a row. Find the probability that (a) the tallest book is at the right end. (b) the tallest and shortest books occupy the end position (c) the tallest and shortest books are together ?

## I would also like to know why you chose that answer.

May 24, 2017

(a): $\frac{1}{5}$ (b): $\frac{1}{10}$ (c): $\frac{2}{5}$

#### Explanation:

There are an overall ${P}_{5}^{5} = 120$ possible cases. To find the probability, divide the number of cases satisfying conditions with the number of maximum possible different situations.

(a):
${P}_{4}^{4} = 24$
Since one of the books have a fixed location, we only need to arrange the rest four books.

(b):
${P}_{2}^{2} \cdot {P}_{3}^{3} = 12$
We take care of the books at the ends and between respectively, and connect the two situations.

(c)
$\left({P}_{2}^{2} \cdot 4\right) \cdot {P}_{3} = 48$
There are ${P}_{2}^{2}$ arrangements for the tallest&shortest books. Also, this tallest-shortest pair has four locations (a.k.a. slots) possible.