Question

From a dessert cart in a restaurant, customers are allowed to pick 3 desserts from the 10 that are displayed. How many combinations are possible?

Answer 1

To determine the number of combinations of desserts that customers can pick from the dessert cart, we can use the combination formula. The formula for combinations is given by:

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

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

In this case, we have:
- \( n = 10 \) (the total number of desserts),
- \( r = 3 \) (the number of desserts to be selected).

Now we can apply the formula:

\[
C(10, 3) = \frac{10!}{3!(10 - 3)!} = \frac{10!}{3! \cdot 7!}
\]

Next, we simplify the expression. We can simplify \( 10! \) as follows:

\[
10! = 10 \times 9 \times 8 \times 7!
\]

Now, substituting this back into the combination formula gives:

\[
C(10, 3) = \frac{10 \times 9 \times 8 \times 7!}{3! \times 7!}
\]

The \( 7! \) in the numerator and the denominator cancel each other out:

\[
C(10, 3) = \frac{10 \times 9 \times 8}{3!}
\]

Next, we calculate \( 3! \):

\[
3! = 3 \times 2 \times 1 = 6
\]

Thus, we can rewrite our combination calculation as:

\[
C(10, 3) = \frac{10 \times 9 \times 8}{6}
\]

Calculating the numerator:

\[
10 \times 9 = 90
\]
\[
90 \times 8 = 720
\]

Now, we divide by \( 6 \):

\[
C(10, 3) = \frac{720}{6} = 120
\]

Thus, the total number of combinations of desserts that customers can choose is:

\[
\boxed{120}
\]