Question
How many ways can a cook arrange 8 or 12 spice jars on a spice rack?
Answers
GPT-4o mini
To determine how many ways a cook can arrange 8 or 12 spice jars on a spice rack, we can utilize the concept of permutations since the arrangement of jars is significant.
1. **Arranging 8 Spice Jars:**
The number of ways to arrange 8 distinct spice jars is given by the factorial of 8, denoted as \(8!\).
\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\]
2. **Arranging 12 Spice Jars:**
Similarly, the number of ways to arrange 12 distinct spice jars is given by the factorial of 12, denoted as \(12!\).
\[
12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600
\]
3. **Total Arrangements:**
To get the total number of arrangements for both 8 and 12 spice jars, we will add the two results together:
\[
\text{Total arrangements} = 8! + 12! = 40320 + 479001600 = 479041920
\]
Thus, the total number of ways the cook can arrange either 8 or 12 spice jars on a spice rack is \( \boxed{479041920} \).
1. **Arranging 8 Spice Jars:**
The number of ways to arrange 8 distinct spice jars is given by the factorial of 8, denoted as \(8!\).
\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\]
2. **Arranging 12 Spice Jars:**
Similarly, the number of ways to arrange 12 distinct spice jars is given by the factorial of 12, denoted as \(12!\).
\[
12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600
\]
3. **Total Arrangements:**
To get the total number of arrangements for both 8 and 12 spice jars, we will add the two results together:
\[
\text{Total arrangements} = 8! + 12! = 40320 + 479001600 = 479041920
\]
Thus, the total number of ways the cook can arrange either 8 or 12 spice jars on a spice rack is \( \boxed{479041920} \).