Eight students are competing for. 1st, 2nd, 3rd, and 4th violin chair in the school's orchestra. How many different ways can all four chairs be filled? (1 point) 4,096 32 1,680 70

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UpStudy AI · Accepted Answer

To determine how many different ways the four chairs can be filled by the eight students, we can use the concept of permutations since the order in which the students are seated matters.

We need to select 4 students from a group of 8 and arrange them in 4 positions (chairs). The formula for permutations is given by:

$$
P(n, r) = \frac{n!}{(n - r)!}
$$

where:
- $$ n $$ is the total number of items (students),
- $$ r $$ is the number of items to choose (chairs),
- $$ ! $$ denotes factorial, which is the product of all positive integers up to that number.

In this case, $$ n = 8 $$ and $$ r = 4 $$.

Now, we can calculate $$ P(8, 4) $$:

$$
P(8, 4) = \frac{8!}{(8 - 4)!} = \frac{8!}{4!}
$$

Calculating $$ 8! $$ and $$ 4! $$:

$$
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
$$
$$
4! = 4 \times 3 \times 2 \times 1
$$

Now, we can simplify $$ P(8, 4) $$:

$$
P(8, 4) = \frac{8 \times 7 \times 6 \times 5 \times 4!}{4!} = 8 \times 7 \times 6 \times 5
$$

Now, let's calculate this step by step:

1. $$ 8 \times 7 = 56 $$
2. $$ 56 \times 6 = 336 $$
3. $$ 336 \times 5 = 1680 $$

Thus, the total number of different ways to fill the four chairs is $$ 1680 $$.

The correct answer is **1,680**.
There are 1,680 different ways to fill the four chairs.

Eight students are competing for. 1st, 2nd, 3rd, and 4th violin chair in the school's orchestra. How many different ways can all four chairs be filled? (1 point) 4,096 32 1,680 70

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