/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 In how many ways can a committee... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 41

In how many ways can a committee of 4 students be formed from a pool of 7 students?

Short Answer

Expert verified
35

Step by step solution

01

Understand the Problem

Determine the number of possible ways to select a committee of 4 students from a pool of 7 students. This is a combination problem because the order of selection does not matter.
02

Identify the Formula for Combinations

Use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of students, and \( r \) is the number of students to be chosen for the committee.
03

Substitute the Given Values into the Formula

For this problem, \( n = 7 \) and \( r = 4 \). Substitute these values into the combination formula: \[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \times 3!} \]
04

Calculate the Factorials

Calculate the factorials: \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \] \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] \[ 3! = 3 \times 2 \times 1 = 6 \]
05

Compute the Combination

Substitute the factorials back into the combination formula: \[ \binom{7}{4} = \frac{5040}{24 \times 6} = \frac{5040}{144} = 35 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factorials
Factorials are a fundamental concept in mathematics, especially in combinatorics. A factorial is the product of all positive integers up to a given number. It's denoted by an exclamation mark, for example, 7 factorial is written as 7!. It's calculated as follows:
7! = 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 5040.
Let's break this down:
  • The factorial of 0 is defined as 1 (0! = 1).
  • The factorial of 1 is 1 (1! = 1).
  • For any number n greater than 1, n! is the product of all integers from 1 to n.
Factorials grow very quickly with larger numbers, making them essential for calculations in permutations and combinations.
combination formula
When we talk about combinations, we refer to selecting items from a larger pool, where the order does not matter. The combination formula is:
\( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
Here:
  • n is the total number of items.
  • r is the number of items to choose.
  • n! is the factorial of n.
  • r! is the factorial of r.
  • (n-r)! is the factorial of the difference between n and r.
For example, if we want to choose 4 students out of 7, we calculate:
\( \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \times 3!} \). This value tells us the number of ways to pick 4 students from a group of 7.
problem-solving steps
Understanding how to approach combination problems step by step is very useful. Let's use the example of forming a committee of 4 from 7 students to illustrate:
  • Step 1: Identify that the problem is a combination, not a permutation since the order does not matter.
  • Step 2: Write down the combination formula:
\( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
  • Step 3: Substitute the values (n = 7, r = 4):
\( \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \times 3!} \)
  • Step 4: Calculate the factorials:

7! = 5040, 4! = 24, and 3! = 6.
  • Step 5: Substitute back into the formula:
\( \binom{7}{4} = \frac{5040}{24 \times 6} = \frac{5040}{144} = 35 \)
This means there are 35 different ways to form the committee. Following these steps ensures you understand each part of the problem and leads you to the correct solution.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Two fair dice are rolled. Determine the probability that the sum of the faces is \(7 .\)

How many different four-letter codes are there if only the letters \(A, B, C, D, E,\) and \(F\) can be used and no letter can be used more than once?

How many arrangements of answers are possible in a multiple-choice test with 5 questions, each of which has 4 possible answers?

The sample space is \(S=\\{1,2,3,4,5,6,\) 7,8,9,10}. Suppose that the outcomes are equally likely. Compute the probability of the event \(E:\) "an even number."

According to the American Pet Products Manufacturers Association's \(2017-2018\) National Pet Owners Survey, there is a \(38 \%\) probability that a U.S. household owns a cat. If a U.S. household is randomly selected, what is the probability that it does not own a cat?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.