4. Probability

Counting

4. Probability

Counting

2:21 minutes

Problem 4.4.23b

Textbook Question

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

b. How many different ways can a committee of five be appointed?

Verified step by step guidance

Step 1: Recognize that the problem involves selecting a committee of 5 members from a pool of 15 candidates. Since the order of selection does not matter, this is a combination problem.

Step 2: Recall the formula for combinations, which is given by:

C (n, r) = \frac{n!}{r! (n - r)!}

, where

n

is the total number of candidates and

r

is the number of members to be selected.

Step 3: Substitute the values into the formula. Here,

n = 15

and

r = 5

. The formula becomes:

C (15, 5) = \frac{15!}{5! (15 - 5)!}

Step 4: Simplify the factorials. Compute

15!

5!

, and

10!

. Then simplify the fraction by canceling out common terms in the numerator and denominator.

Step 5: Perform the division to find the total number of combinations. This will give the total number of ways to select a committee of 5 members from 15 candidates.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this scenario, we need to choose 5 members from 15 candidates, which is a classic example of a combination problem. The formula for combinations is given by 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.

Factorial

Factorial is a mathematical operation that multiplies a number by all positive integers less than it. It is denoted by n! and is essential in calculating combinations and permutations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is crucial for solving problems involving arrangements and selections in statistics.

Counting Principles

Counting principles are fundamental rules used to determine the number of ways to arrange or select items. In this context, the combination formula is applied to count the different ways to form a committee from a pool of candidates. Mastery of counting principles allows for efficient problem-solving in various statistical scenarios, especially when dealing with large sets of data.

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