Answer the questions below.(If necessary, consult a list of formulas.)(a) There are 16 appetizers available at a restaurant. From these, Abdul is to choose 12 for his party. How many groups of 12 appetizers are possible?(b) A certain committee consists of 16 people. From the committee, a president, a vice-president, and a treasurer are to be chosen. In how many ways can these 3 office
Question
Answer the questions below.(If necessary, consult a list of formulas.)(a) There are 16 appetizers available at a restaurant. From these, Abdul is to choose 12 for his party. How many groups of 12 appetizers are possible?(b) A certain committee consists of 16 people. From the committee, a president, a vice-president, and a treasurer are to be chosen. In how many ways can these 3 office
Solution
(a) The number of ways to choose 12 appetizers from 16 is given by the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of items, k is the number of items to choose, and "!" denotes factorial.
So, C(16, 12) = 16! / [12!(16-12)!] = 16! / [12!4!] = (16151413) / (4321) = 1820.
So, there are 1820 possible groups of 12 appetizers that Abdul can choose for his party.
(b) The number of ways to choose a president, a vice-president, and a treasurer from a committee of 16 people is given by the permutation formula P(n, k) = n! / (n-k)!, where n is the total number of items, k is the number of items to choose, and "!" denotes factorial.
So, P(16, 3) = 16! / (16-3)! = 16! / 13! = 161514 = 3360.
So, there are 3360 ways to choose these 3 officers from the committee.
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