A corporation must appoint a president, chief executive officer(CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with three different members. There are 16 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below.a.) How many different ways can the officers be appointed?There are different ways to appoint the officers.b.) How many different ways can the committee be appointed?There are different ways to appoint the committee.c.) What is the probability of randomly selecting the committee members and getting the three youngest of the qualified candidates? (Type an integer or a simplified fraction.)
Question
A corporation must appoint a president, chief executive officer(CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with three different members. There are 16 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below.a.) How many different ways can the officers be appointed?There are different ways to appoint the officers.b.) How many different ways can the committee be appointed?There are different ways to appoint the committee.c.) What is the probability of randomly selecting the committee members and getting the three youngest of the qualified candidates? (Type an integer or a simplified fraction.)
Solution
a.) The number of different ways the officers can be appointed is calculated by the permutation formula, which is nPr = n! / (n-r)!. Here, n is the total number of candidates (16) and r is the number of positions to fill (4). So, the calculation would be 16P4 = 16! / (16-4)! = 43680 different ways to appoint the officers.
b.) The number of different ways the committee can be appointed is calculated by the combination formula, which is nCr = n! / r!(n-r)!. Here, n is the total number of candidates (16) and r is the number of positions to fill (3). So, the calculation would be 16C3 = 16! / 3!(16-3)! = 560 different ways to appoint the committee.
c.) The probability of randomly selecting the committee members and getting the three youngest of the qualified candidates is calculated by dividing the number of favorable outcomes by the total number of outcomes. Here, the favorable outcome is 1 (selecting the three youngest candidates) and the total number of outcomes is the number of ways to select 3 members from 16, which we calculated in part b as 560. So, the probability would be 1/560.
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