Here, the factorials you mention don't have anything to do with probability - rather, they have to do with combinatorics. This is because in this context, you are talking about uniformly random elements of some set - and so counting up the elements with a specific property is a necessary step of the process.
Factorials are important because 
is the number of ways to list - in order - a set of 
objects that are distinguishable. Because of this, it also comes up in other arrangements - such as the number of ways to choose 
elements from a set of 
(in an order or otherwise). Further, many other combinatorial constructions can be carried out starting with these basic ideas of arrangements, and so they involve factorials as well.
You are absolutely right that distinguishable is a keyword that screams factorial; however, distinguishability of the objects is often implicit in the problem, rather than explicitly stated. The big thing: look for any situation in which you are arranging objects in some way.
Answer from Nick Peterson on Stack ExchangeDLsun
dlsun.github.io › probability › factorial.html
Lesson 2 The Factorial | Introduction to Probability
So there are \[ 52 \cdot 51 \cdot 50 \cdot \ldots \cdot 2 \cdot 1 \] ways to arrange the 52 cards in a deck. This is such an important quantity in probability and counting that it has been given a special name. Definition 2.1 (Factorial) The quantity \(n!\) (pronounced: “n factorial”) is ...
Videos
Probability: Intro to Factorials
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Factorial and counting seat arrangements | Probability and Statistics ...
Probability using combinations (video)
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Permutations Combinations Factorials & Probability - YouTube
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Probability - factorial notation - YouTube
Statlect
statlect.com › glossary › factorial
Factorial | Use in probability and statistics
The number of possible permutations is equal to . Example Consider the first three letters of the alphabet: . There are ways of ordering these letters: A combination is a way of selecting objects from a list of . The order of selection does not matter and each object can be selected only once. ... A partition is a way of subdividing objects into groups having numerosities . ... Factorials have numerous important applications in the analysis of probability distributions.
Statistics How To
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Factorial: Simple Definition, Examples & Distribution - Statistics How To
June 11, 2024 - Factorials (!) are products of whole numbers up to the number of interest. For example, 3! (read “three factorial”) equals 3 * 2 * 1 = 6. The equation P(x) = ΣP(x | y) * P(y) states that the total probability of event x happening is equal ...
Quora
quora.com › How-do-factorials-tell-you-the-total-number-of-outcomes
How do factorials tell you the total number of outcomes? - Quora
Answer (1 of 5): Factorials are more or less shorthand. 4! = 4 x 3 x 2 x 1 if you had 4 scrabble letters A, B, C and D then an organised way of working out the possible combinations would be to imagine the 4 adjacent squares they could be placed on. Looking at the first square, there are 4 ways ...
Reddit
reddit.com › r/askscience › why do we use factorial to get possible combinations in the card deck?
r/askscience on Reddit: Why do we use factorial to get possible combinations in the card deck?
December 1, 2015 -
I saw this famous fact in some thead on reddit that there are less visible stars than there are possible combinations of outcomes when shuffling a deck of 52 cards.
That is by using factorial. And I've been taught that x! or "factorial" is an arithmetic process used only when elements of the group can repeat themselves, i.e. your outcome could be a deck full of aces. But this outcome is impossible.
If this is wrong, does this mean that there is a different proces than factorial that gives you even larger number?
Top answer 1 of 18
529
You are thinking of a slightly different problem. If you draw a new card at random from the deck and put it back 52 times in a row, then you would have 52 possibilities for the first card, 52 possibilities for the second card, etc. and the number of possible "shuffles" you could make would be: 52 * 52 * 52... = 5252 = 1.7e89. In that scenario you could pick up 52 aces of hearts in a row, though with a probability of 1/1.7e89. But shuffling a deck isn't like that. Shuffling a deck is equivalent to starting with an unshuffled deck and picking cards at random from it but not replacing them after you take them out. There are 52 outcomes for the first card. For each possible first card, there are only 51 cards to choose next. And for each of those combinations, there are only 50 possible number three cards to choose. So the total number of shuffles is 52 * 51 * 50 *... = 52! = 8.1e67. Still a big number, but a much smaller number than the first one we calculated. When you shuffle a deck it is obviously impossible to get 52 aces of hearts in a row, unless you are playing poker with someone who is both very sketchy and not too bright.
2 of 18
433
I think you're overthinking it. Suppose you have 52 cards scattered about. You decide to pick one of the 52 cards and lay it down in front of you - that's the bottom card of the deck. Now, you pick one of the 51 remaining cards and place it on top. Then, pick one of the remaining 50 cards and place it on top, etc. At each step, you have 52 choices, and then 51 choices, etc, so the product of these number of choices at each step is your number of possible decks: 52!
Steemit
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What Is A Factorial and How Are Factorials Used In Statistics? — Steemit
April 4, 2018 - Factorial functions are used throughout statistics and probability. For example, the number of ways that n articles may be arranged is n!, or to calculate the odds of winning the UK national Lottery, by picking six numbers from forty-nine, the odds are one in
CK-12 Foundation
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Factorials and Combinations ( Read ) | Probability | CK-12 Foundation
To better organize out content, we have unpublished this concept. This page will be removed in future. ... A review of factorials and combinations.
CK-12 Foundation
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Story of Mathematics
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Factorial - Explanation & Examples
October 26, 2023 - Or if we want to select a team of $5$ people from $10$ available members, how many different teams can we make? Factorial is a mathematical operation that helps us in figuring out such arrangements and hence plays an important role in probability ...
Assignment Expert
assignmentexpert.com › home › how to use factorials in probability calculations
Use of Factorials in Probability Calculations
November 30, 2020 - For example, 3!=3 × 2 × 1=6. Factorial from 4 is considered as the 4 × 3 × 2 × 1, that is 24. And so on. It seems a quite easy calculation, but only for the first few numbers as five factorial is 120, 6!= 5040 and 7! is already 40320 (for understanding their correct evaluation to check the number of allowable permutations for these numbers ).
Shmoop
shmoop.com › probability-statistics › factorials-permutations.html
Probability and Statistics Factorials and Permutations
Just one of the factorials of life. The number (n – r) is the number of objects we'll have left over after we fill all available spaces. Also, n! is the number of permutations if we use all n objects; dividing by (n – r)! accounts for the spaces we can't fill because we don't have them to begin with. We should be getting them any day, though. Amazon has them on back-order. We abbreviate "the number of permutations possible with n objects and r places" in symbols by
Williams College
web.williams.edu › Mathematics › sjmiller › public_html › math › talks › IntroProbability.pdf pdf
Introduction to Probability Steven J Miller, Williams College sjm1@Williams.edu
The factorial function is great for ordering n objects; what if we only want to · order some of them? nPr or nPr is the number of ways to choose r people from n ... In a hand of bridge, each of the four players is dealt 13 cards.
Reddit
reddit.com › r/learnmath › combinatorics: when to use exponents vs. factorials
r/learnmath on Reddit: Combinatorics: When to use exponents vs. factorials
January 26, 2023 -
Is there are general rule or is there a way I can tell a problem needs to be either solved by using factorials (multiplication) or exponents?
Top answer 1 of 2
3
No, there's really no especially general rule. TBH, it doesn't make much sense to draw a sharp line between "exponent problems" and "factorial problems" since a given problem might end up needing both or neither. The closest I can really get to a generalization (although it's one that's really only relevant for certain sorts of basic problems) is that factorials show up when you're counting ordered sequences with no repetition (i.e. permutations), while exponentiation shows up when you're counting ordered sequences with repetition allowed. E.g. the number of ways to form a 2-letter word using the letters A, B, and C is 3 * 2 = 6 if you don't allow repeated letters, while it's 3 * 3 = 9 if you allow repetition. But this distinction breaks down pretty quickly once the problems get even slightly more complicated, e.g. factorials show up in the problem of counting unordered collections without repetition (i.e. subsets) as well (though to be fair the reason they show up has to do in large part with permutations). So really, I would advise just thinking about each problem individually instead of trying to pattern-match it as an "exponent problem" or "factorial problem".
2 of 2
2
If every choice is from the same pool, exponents. If the options decrease with every choice, factorials
Reddit
reddit.com › r/learnmath › when do you actually use factorials in real life? not sure why i learned it because it’s never been applicable for me.
r/learnmath on Reddit: When do you actually use factorials in real life? Not sure why I learned it because it’s never been applicable for me.
December 27, 2020 - Depending on what you do for a career they could be quite applicable. If you have to know how many ways there are to arrange things for instance, factorials come up in the calculation (unless you want to write out 1 x 2 x 3 x 4 x ...).
YouTube
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When To Use Factorial In Probability? - The Friendly Statistician - YouTube
When To Use Factorial In Probability? Understanding the role of factorials in probability can greatly enhance your ability to tackle various mathematical cha...
Published January 25, 2025
Wikipedia
en.wikipedia.org › wiki › Factorial
Factorial - Wikipedia
1 week ago - The earliest uses of the factorial function involve counting permutations: there are ... {\displaystyle n} distinct objects into a sequence. Factorials appear more broadly in many formulas in combinatorics, to account for different orderings of objects.
Penn State Statistics
online.stat.psu.edu › statprogram › reviews › algebra › factorials
A.3 Factorials | STAT ONLINE
A factorial is a mathematical operation in which you multiply the given number by all of the positive whole numbers less than it. In other words \(n!=n \times (n-1) \times …
Narkive
mathematics.science.narkive.com › KgDba4mx › how-do-u-do-factorials-and-probability
how do u do factorials and probability?
Examples: 5! = 5 X 4 X 3 X 2 X 1 = 120 4! = 4 X 3 X 2 X 1 = 24 3! = 3 X 2 X 1 = 6 Calculating combinations is based on factorials. You will learn about combinations later. Combinations allow you to calculate things like the likelihood (probability) of being dealt a specific hand in a card game. If you wanted to know the likelihood of rolling two fair dice and getting snake eyes, you can use probability.
Statistics LibreTexts
stats.libretexts.org › bookshelves › introductory statistics › support course for elementary statistics › operations on numbers
Factorials and Combination Notation - Statistics LibreTexts
April 9, 2022 - When you do use technology, you should get: ... One of the most important applications of factorials is combinations which count the number of ways of selecting a smaller collection from a larger collection when order is not important.