To find the number of ways to choose 3 books from a shelf holding 5 different books, we can use the combination formula, which is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
In this case, \( n = 5 \) (the total number of books) and \( r = 3 \) (the number of books to choose). Plugging in these values, we get:
\[ \binom{5}{3} = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! \cdot 2!} \]
Now calculating the factorials:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] \[ 3! = 3 \times 2 \times 1 = 6 \] \[ 2! = 2 \times 1 = 2 \]
Now substituting these values into the combination formula:
\[ \binom{5}{3} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 \]
Thus, the number of ways to choose 3 books from a shelf holding 5 different books is \( \boxed{10} \).