1. **Problem:** How many combinations of 6 objects are there, taken 3 at a time? 2. **Formula:** The number of combinations of $n$ objects taken $r$ at a time is given by the binomial coefficient: $$ C(n,r) = \frac{n!}{r!(n-r)!} $$ where $n!$ denotes the factorial of $n$. 3. **Apply the formula:** Here, $n=6$ and $r=3$. $$ C(6,3) = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} $$ 4. **Calculate factorials:** $$ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 $$ $$ 3! = 3 \times 2 \times 1 = 6 $$ 5. **Substitute values:** $$ C(6,3) = \frac{720}{6 \times 6} = \frac{720}{36} = 20 $$ 6. **Answer:** There are **20** combinations of 6 objects taken 3 at a time.