1. **Problem statement:** We need to find the number of ways a basketball coach can select his first 5 players from a 15-man basketball team. 2. **Formula used:** This is a combination problem because the order of selection does not matter. The formula for combinations is: $$ C(n, r) = \frac{n!}{r!(n-r)!} $$ where $n$ is the total number of players, and $r$ is the number of players to select. 3. **Apply the formula:** Here, $n=15$ and $r=5$. $$ C(15, 5) = \frac{15!}{5!(15-5)!} = \frac{15!}{5!10!} $$ 4. **Simplify the factorial expression:** $$ \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10!}{5! \times 10!} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5!} $$ 5. **Calculate $5!$:** $$ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $$ 6. **Calculate numerator:** $$ 15 \times 14 = 210 $$ $$ 210 \times 13 = 2730 $$ $$ 2730 \times 12 = 32760 $$ $$ 32760 \times 11 = 360360 $$ 7. **Divide numerator by denominator:** $$ \frac{360360}{120} = 3003 $$ **Final answer:** There are $3003$ ways for the coach to select his first 5 players from the 15-man team.