Question

How many flower arrangements can be made with 10 roses and 12 daisies if each arrangement has 2 roses and 3 daisies?

Answer 1

To determine how many flower arrangements can be made with 10 roses and 12 daisies, where each arrangement consists of 2 roses and 3 daisies, we need to calculate the number of ways to choose the roses and the daisies separately and then multiply those results.

**Step 1: Choose 2 roses from 10.**

The number of ways to choose 2 roses from a total of 10 can be calculated using the combination formula:

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

In this case, \( n = 10 \) (the total number of roses) and \( r = 2 \) (the number of roses to choose):

\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
\]

**Step 2: Choose 3 daisies from 12.**

Next, we calculate the number of ways to choose 3 daisies from a total of 12 using the same combination formula:

Here, \( n = 12 \) (the total number of daisies) and \( r = 3 \) (the number of daisies to choose):

\[
\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = \frac{1320}{6} = 220
\]

**Step 3: Calculate the total number of arrangements.**

Now that we have the number of ways to choose the roses and the daisies, we multiply these results to find the total number of arrangements:

\[
\text{Total arrangements} = \binom{10}{2} \times \binom{12}{3} = 45 \times 220
\]

Calculating the final result:

\[
45 \times 220 = 9900
\]

Thus, the total number of flower arrangements that can be made with 10 roses and 12 daisies, selecting 2 roses and 3 daisies for each arrangement, is

\[
\boxed{9900}
\]