There are
17
appetizers available at a restaurant. From these, Kim is to choose
12
for her party. How many groups of
12
appetizers are possible
To find out how many groups of 12 appetizers are possible from the 17 available, we can use the combination formula. The combination formula is given by:
C(n, r) = n! / (r!(n-r)!)
Where:
- n is the total number of items (17 in this case)
- r is the number of items to be chosen (12 in this case)
- n! denotes the factorial of n
So, substituting the values into the formula, we have:
C(17, 12) = 17! / (12!(17-12)!)
Now, let's calculate the value using this formula:
17! = 17 x 16 x 15 x 14 x 13 x 12!
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Plugging these values into the formula:
C(17, 12) = (17 x 16 x 15 x 14 x 13 x 12!) / (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 5!)
Calculating this expression, we find:
C(17, 12) = (17 x 16 x 15 x 14 x 13) / (12 x 11 x 10 x 9 x 8 x 7 x 6)
Simplifying the expression:
C(17, 12) = 6188
Therefore, there are 6,188 possible groups of 12 appetizers that can be chosen from the 17 available.
To find the number of groups of 12 appetizers that Kim can choose from the 17 available options, we use the concept of combinations.
The formula for combinations is given by:
nCr = n! / (r! * (n - r)!)
Where n is the total number of options (17 in this case) and r is the number of options to be chosen (12 in this case).
Using the formula, we can calculate the number of groups of 12 appetizers as follows:
17C12 = 17! / (12! * (17 - 12)!)
Calculating the factorial values, we have:
17! = 17 * 16 * 15 * 14 * 13 * 12!
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
(17 - 12)! = 5!
Substituting these values into the formula, we get:
17C12 = (17 * 16 * 15 * 14 * 13 * 12!) / (12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 * 5!)
Simplifying the expression, we get:
17C12 = (17 * 16 * 15 * 14 * 13) / (5 * 4 * 3 * 2 * 1)
Finding the product of the numbers in the numerator and the denominator, we have:
17C12 = 436,590 / 120
Dividing the numerator by the denominator, we get:
17C12 = 3,638
Therefore, there are 3,638 groups of 12 appetizers that Kim can choose from.