Ex: You have five books and you are going to put three on a bookshelf. How many different ways can the books be ordered on the bookshelf? Counting Rule 5: Combinations:The number of ways of selecting X objects from n objects, irrespective of order, is (}_{n)C_(x)=(n!)/(X!(n-X)!)

To find the number of different ways to arrange three books out of five on a bookshelf, we can use the formula for combinations:<br /><br />${}_{n}C_{x}=\frac {n!}{X!(n-X)!}$<br /><br />In this case, n = 5 (the total number of books) and x = 3 (the number of books we want to arrange on the bookshelf).<br /><br />Plugging in the values, we get:<br /><br />${}_{5}C_{3}=\frac {5!}{3!(5-3)!}$<br /><br />Simplifying the expression, we have:<br /><br />${}_{5}C_{3}=\frac {5!}{3!2!}$<br /><br />Calculating the factorials, we get:<br /><br />${}_{5}C_{3}=\frac {5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(2 \times 1)}$<br /><br />Simplifying further, we have:<br /><br />${}_{5}C_{3}=\frac {5 \times 4}{2 \times 1}$<br /><br />Finally, calculating the expression, we get:<br /><br />${}_{5}C_{3}=10$<br /><br />Therefore, there are 10 different ways to arrange three books out of five on a bookshelf.