Twenty people are to be seated at 3 Circular restaurant tables that seat 8,7, and 5 people respectively. Determine the number of ways that this can be done. Express your answer in scientific notation to two decimal place.
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To solve this problem, we'll use a basic principle of combinatorics called permutations. Permutations refers to the arrangement of objects in a specific order.
Let's start by determining the number of ways the 20 people can be seated at each table separately.
For the first table that seats 8 people, the number of ways to arrange the 20 people is denoted as 20P8, which can be calculated as:
20P8 = 20! / (20-8)!
= 20! / 12!
= (20 × 19 × 18 × 17 × 16 × 15 × 14 × 13)
Similarly, for the second table that seats 7 people, the number of ways is 20P7:
20P7 = 20! / (20-7)!
= 20! / 13!
= (20 × 19 × 18 × 17 × 16 × 15 × 14)
Lastly, for the third table that seats 5 people, the number of ways is 20P5:
20P5 = 20! / (20-5)!
= 20! / 15!
= (20 × 19 × 18 × 17 × 16)
Now, to find the number of total ways the 20 people can be seated at all three tables together, we multiply the number of ways for each table:
Total ways = 20P8 × 20P7 × 20P5
Plugging in the calculated values, we get:
Total ways = (20 × 19 × 18 × 17 × 16 × 15 × 14 × 13) × (20 × 19 × 18 × 17 × 16 × 15 × 14) × (20 × 19 × 18 × 17 × 16)
Now, we can calculate this expression to get the final answer:
Total ways = (20 × 19 × 18 × 17 × 16 × 15 × 14 × 13) × (20 × 19 × 18 × 17 × 16 × 15 × 14) × (20 × 19 × 18 × 17 × 16) = 8.465237 × 10^23
Therefore, the number of ways that 20 people can be seated at the three circular restaurant tables is approximately 8.47 × 10^23.