Questions: President Joseph R. Biden's cabinet consisted of the Secretaries of Agriculture, Commerce, Defense, Education, Energy, Health and Human Services, Homeland Security, Housing and Urban Development, Interior, Labor, State, Transportation, Treasury, Veterans Affairs, and the Attorney General. Assuming that President Biden had 28 candidates, including Dr. Janet Yellen, to fill these posts (and wished to assign no one to more than one post), complete the following sentences. (a) The sample space is the set of all lists of people chosen from . (b) The event that Dr. Janet Yellen is the Secretary of the Treasury is the set of all lists of 15 people chosen from , in which Dr. Janet Yellen occupies the position.

President Joseph R. Biden's cabinet consisted of the Secretaries of Agriculture, Commerce, Defense, Education, Energy, Health and Human Services, Homeland Security, Housing and Urban Development, Interior, Labor, State, Transportation, Treasury, Veterans Affairs, and the Attorney General. Assuming that President Biden had 28 candidates, including Dr. Janet Yellen, to fill these posts (and wished to assign no one to more than one post), complete the following sentences.
(a) The sample space is the set of all lists of people chosen from .
(b) The event that Dr. Janet Yellen is the Secretary of the Treasury is the set of all lists of 15 people chosen from , in which Dr. Janet Yellen occupies the position.

Transcript text: President Joseph R. Biden's cabinet consisted of the Secretaries of Agriculture, Commerce, Defense, Education, Energy, Health and Human Services, Homeland Security, Housing and Urban Development, Interior, Labor, State, Transportation, Treasury, Veterans Affairs, and the Attorney General. Assuming that President Biden had 28 candidates, including Dr. Janet Yellen, to fill these posts (and wished to assign no one to more than one post), complete the following sentences. (a) The sample space is the set of all lists of people chosen from . (b) The event that Dr. Janet Yellen is the Secretary of the Treasury is the set of all lists of 15 people chosen from , in which Dr. Janet Yellen occupies the position.

Solution

Solution Steps

To solve this problem, we need to understand the concepts of permutations and combinations. Here’s a high-level approach to solve the given parts:

(a) The sample space is the set of all possible ways to choose 15 people from 28 candidates, where the order of selection matters (since each person is assigned a unique position).

(b) The event that Dr. Janet Yellen is the Secretary of the Treasury is a subset of the sample space where Dr. Janet Yellen is fixed in the position of Secretary of the Treasury, and the remaining 14 positions are filled by choosing 14 people from the remaining 27 candidates.

Solution Approach

Calculate the total number of permutations for choosing 15 people from 28 candidates.
Calculate the number of permutations where Dr. Janet Yellen is fixed in the Secretary of the Treasury position, and the remaining 14 positions are filled from the remaining 27 candidates.

Step 1: Calculate the Total Number of Permutations

To find the total number of ways to choose 15 people from 28 candidates, where the order of selection matters, we use the permutation formula:

\[ P(n, k) = \frac{n!}{(n-k)!} \]

For \( n = 28 \) and \( k = 15 \):

\[ P(28, 15) = \frac{28!}{(28-15)!} = 48,962,152,914,554,880,000 \]

Step 2: Calculate the Number of Permutations with Dr. Janet Yellen as Secretary of the Treasury

To find the number of ways to choose 14 people from the remaining 27 candidates (since Dr. Janet Yellen is fixed in one position), we use the permutation formula again:

\[ P(n, k) = \frac{n!}{(n-k)!} \]

For \( n = 27 \) and \( k = 14 \):

\[ P(27, 14) = \frac{27!}{(27-14)!} = 1,748,648,318,376,960,000 \]