Question: Ms. Lam has to arrange her jazz choir into rows. The first row has spots for 3 sopranos, and there are a total of 10 sopranos in the choir. How many ways can Ms. Lam arrange the sopranos in the first row?
1. **State the problem:** We need to find how many ways Ms. Lam can arrange 3 sopranos out of 10 in the first row.
2. **Formula used:** Since the order of arrangement matters (arranging in a row), we use permutations. The number of permutations of choosing $k$ elements from $n$ is given by:
$$ P(n,k) = \frac{n!}{(n-k)!} $$
3. **Apply the formula:** Here, $n=10$ and $k=3$, so
$$ P(10,3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} $$
4. **Calculate factorials:**
$$ 10! = 10 \times 9 \times 8 \times 7! $$
So,
$$ P(10,3) = \frac{10 \times 9 \times 8 \times 7!}{7!} $$
5. **Cancel common factors:**
$$ P(10,3) = 10 \times 9 \times 8 $$
6. **Multiply:**
$$ 10 \times 9 = 90 $$
$$ 90 \times 8 = 720 $$
7. **Final answer:**
There are **720 ways** Ms. Lam can arrange the 3 sopranos in the first row.