The total number of ways to choose 4 marbles from 16 is: - Sourci
The Total Number of Ways to Choose 4 Marbles from 16: The Science Behind Combinations
The Total Number of Ways to Choose 4 Marbles from 16: The Science Behind Combinations
When faced with the question — “How many ways can you choose 4 marbles from a set of 16?” — many might initially think in terms of simple counting, but the true elegance lies in combinatorics. This article explores the exact mathematical answer, explains the concept of combinations, and reveals how you calculate the total number of ways to select 4 marbles from 16.
Understanding the Context
Understanding the Problem
At first glance, choosing 4 marbles from 16 might seem like a straightforward arithmetic problem. However, the key distinction lies in whether the order of selection matters:
- If order matters, you’re dealing with permutations — calculating how many ways marbles can be arranged when position is important.
- But if order doesn’t matter, and you only care about which marbles are selected (not the sequence), you’re looking at combinations.
Since selecting marbles for a collection typically concerns selection without regard to order, we focus on combinations — specifically, the number of combinations of 16 marbles taken 4 at a time, denoted mathematically as:
Image Gallery
Key Insights
$$
\binom{16}{4}
$$
What is a Combination?
A combination is a way of selecting items from a larger set where the order of selection is irrelevant. The formula to compute combinations is:
$$
\binom{n}{r} = \frac{n!}{r!(n - r)!}
$$
🔗 Related Articles You Might Like:
📰 cast of call the midwife 📰 serie cbs 📰 cast of y&r soap 📰 Best Roblox External 📰 You Wont Believe These Savage Scene Splitting Moments In Rocky Horror 4358698 📰 Why Every Investor In Indianapolis Should Know Fidelity Investments Indo 5563488 📰 What Is Bcc In Email 📰 This Extension Ladder Transforms Your Diy Projectsexperience Strength Like Never Before 3326668 📰 Nyseamerican Sens 📰 Mat Donovan 435951 📰 Big Announcement The Roots Albums And The World Is Watching 📰 Shock Moment Object Push Profile And The Evidence Appears 📰 Wordle Hiunt 9557431 📰 Best Amazon Prime Shows 1356975 📰 Jordan Sweat 4168856 📰 Is Sonic Racing Crossworlds Crossplay 📰 Verizon Headquarters 📰 Azure Sql Server Kill SessionFinal Thoughts
Where:
- $ n $ = total number of items (here, 16 marbles)
- $ r $ = number of items to choose (here, 4 marbles)
- $ ! $ denotes factorial — the product of all positive integers up to that number (e.g., $ 5! = 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 120 $)
Using this formula:
$$
\binom{16}{4} = \frac{16!}{4!(16 - 4)!} = \frac{16!}{4! \cdot 12!}
$$
Note that $ 16! = 16 \ imes 15 \ imes 14 \ imes 13 \ imes 12! $, so the $ 12! $ cancels out:
$$
\binom{16}{4} = \frac{16 \ imes 15 \ imes 14 \ imes 13}{4 \ imes 3 \ imes 2 \ imes 1}
$$
Calculating the Value
Now compute the numerator and denominator:
-
Numerator:
$ 16 \ imes 15 = 240 $
$ 240 \ imes 14 = 3,360 $
$ 3,360 \ imes 13 = 43,680 $ -
Denominator:
$ 4 \ imes 3 = 12 $, $ 12 \ imes 2 = 24 $, $ 24 \ imes 1 = 24 $
