Question

How many different ways are there to choose 5 books from a shelf of 24 books if the order in which they are selected doesn't matter?(1 point)
Responses

5,100,480
5,100,480

42,504
42,504

7

1
3
7

1
3

22
22
Question 2
How many ways are there if the order does matter?(1 point)
Responses

7

1
3
7

1
3

22
22

5,100,480
5,100,480

42,504

Answer 1

To determine the number of different ways to choose 5 books from a shelf of 24 books, we can use combinations for the first question since the order does not matter.

### Question 1: Order Does Not Matter
The formula for combinations is given by:

\[
C(n, r) = \frac{n!}{r!(n-r)!}
\]

where \( n \) is the total number of items (books) to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial.

For this question:
- \( n = 24 \)
- \( r = 5 \)

Thus, we calculate:

\[
C(24, 5) = \frac{24!}{5!(24-5)!} = \frac{24!}{5! \cdot 19!}
\]

Calculating this step-by-step:

First, we simplify \( 24! \) by expressing it in terms of \( 19! \):

\[
C(24, 5) = \frac{24 \times 23 \times 22 \times 21 \times 20}{5 \times 4 \times 3 \times 2 \times 1}
\]

Calculating the numerator:

\[
24 \times 23 = 552
\]
\[
552 \times 22 = 12144
\]
\[
12144 \times 21 = 255024
\]
\[
255024 \times 20 = 5100480
\]

Now calculating the denominator:

\[
5! = 120
\]

Now, we divide the results:

\[
C(24, 5) = \frac{5100480}{120} = 42504
\]

So, the answer for Question 1 is **42,504**.

### Question 2: Order Does Matter
If the order does matter, we use permutations instead. The formula for permutations is given by:

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

where \( n \) is the total number of items, and \( r \) is the number of items to arrange.

Thus, for this question:

\[
P(24, 5) = \frac{24!}{(24-5)!} = \frac{24!}{19!}
\]

Simplifying this, we have:

\[
P(24, 5) = 24 \times 23 \times 22 \times 21 \times 20
\]

Calculating the above:

\[
24 \times 23 = 552
\]
\[
552 \times 22 = 12144
\]
\[
12144 \times 21 = 255024
\]
\[
255024 \times 20 = 5100480
\]

Thus, the total number of ways to arrange 5 books from 24 if order does matter is **5,100,480**.

So, for Question 2, the answer is **5,100,480**.