The factorial function (article) | Khan Academy (2024)

Want to join the conversation?

Log in

  • Anne

    9 years agoPosted 9 years ago. Direct link to Anne's post “Can someone clarify why 0...”

    Can someone clarify why 0! = 1? I'm really getting lost on the last paragraph, specifically:

    "It's the product of all integers greater than or equal to 1 and less than or equal to 0. But there are no such integers. Therefore, we define 0! to equal the identity for multiplication, which is 1."

    (33 votes)

    • Cameron

      9 years agoPosted 9 years ago. Direct link to Cameron's post “n! is the product of all ...”

      The factorial function (article) | Khan Academy (4)

      The factorial function (article) | Khan Academy (5)

      The factorial function (article) | Khan Academy (6)

      n! is the product of all integers greater than or equal to 1 and less than or equal to n
      or more simply:
      the product of all integers from 1 to n (starting at 1 and going up to n)
      i.e. 1 * 2 * 3 * 4 * 5 * ... * n-1 * n

      so when n=0 we have
      the product of all integers from 1 to 0 (starting at 1 and going up to 0)
      But 1 is already larger than 0 so there are no integers from 1 to 0 (counting up)
      So what do we use for the product of no integers ?
      We use 1.
      Why ?
      Because it is convenient, it makes makes math easier, and most importantly it makes our formulas that could calculate 0! make sense. The article gives an example of how it makes our formulas make sense. It shows that when we calculate the number of ways of choosing a group of n items from n items (which should obviously be 1), the formula needs 0! to be 1.

      Hope this makes sense

      (81 votes)

  • Maddie Walker

    7 years agoPosted 7 years ago. Direct link to Maddie Walker's post “I'm still really confused...”

    I'm still really confused on this subject. If one of y'all could explain how n and n! work in a really simple almost kindergarten version, that would be amazing! I'm trying to relate this so something that I have already learned/know about so the concept will be easily identified and understandable. Thanks in advance!

    (9 votes)

    • Cameron

      7 years agoPosted 7 years ago. Direct link to Cameron's post “n! is the product (multip...”

      The factorial function (article) | Khan Academy (10)

      n! is the product (multiplication) of all the whole numbers from n to 1
      e.g.
      3! = 3 * 2 * 1
      2! = 2 * 1
      1! = 1

      0! is a special case. We say that 0! = 1.

      We use factorials when we look at permutations and combinations.
      Permutations tell us how many different ways we can arrange things if their order matters.
      Combinations tells us how many ways we can choose k item from n items if their order does not matter.

      If we have n different items, then we can permute (arrange) them in n! different arrangements.
      e.g.
      If we have 3 items: A,B,C then there are 3! = 3 * 2 * 1 = 6 different permuations
      Here they are:
      ABC, ACB, BAC, BCA, CAB, CBA

      If we have n items there are n!/(k! * (n-k)!) different combinations of choosing k items from those n items.
      e.g.
      If we 5 items: A,B,C,D,E and we want to choose 2 items at a time then there are:
      5!/( 2! * (5-2)!) = 5! / (2! * 3!) = 120/(2 * 6) = 10 different combinations
      Here they are:
      AB, AC, AD, AE, BC, BD, BE, CD, CE, DE
      Note: CB would be considered as the same combination as BC, because order doesn't matter

      For more info check out:
      https://www.khanacademy.org/math/precalculus/prob-comb/combinatorics-precalc/v/factorial-and-counting-seat-arrangements
      https://www.khanacademy.org/math/precalculus/prob-comb/combinations/v/introduction-to-combinations

      (24 votes)

  • lauren.finkle

    9 years agoPosted 9 years ago. Direct link to lauren.finkle's post “In the for loop when we s...”

    In the for loop when we say result = result * i; how do these two results not override each other? How does the computer know that one result is referring to the previous sum of the multiplications and the other is that value times i?

    (2 votes)

    • jdsutton

      9 years agoPosted 9 years ago. Direct link to jdsutton's post “`result = result * i;` is...”

      The factorial function (article) | Khan Academy (14)

      result = result * i; is really telling the computer to do this:
      1. Compute the value of result * i and store it somewhere
      2. Take the stored value and assign result to that value

      When you use the = operator, the entire right hand side is evaluated to some value, and then that value is put into the left hand side variable. Everything is happening in steps, not all at once.

      (12 votes)

  • Vu Tran

    9 years agoPosted 9 years ago. Direct link to Vu Tran's post “In the example given abov...”

    In the example given above where you have n t-shirts and want to be able to pack only k amount, the given formula is n!/(k!⋅(n−k)!)

    In the permutations course given in the link, the formula is stated to to be n!/(n-k)!

    Can someone explain where the extra k! in the denominator came from?

    (3 votes)

    • Cameron

      9 years agoPosted 9 years ago. Direct link to Cameron's post “n!/(k! (n-k)!) would tell...”

      n!/(k! (n-k)!) would tell you the number of ways you could pick k t-shirts from n shirts where the order does not matter

      n!/(n-k)! would tell you the number of ways you could pick k t-shirts from n shirts where the order matters e.g. the first shirt you pick you write 1 on it, the 2nd shirt you write 2 on it, etc.

      Dividing by k! accounts for the number of ways each selection of k shirts will be counted.
      e.g. Suppose k= 3 and one possible selection of shirts is shirts: 5, 7 and 13
      You could have picked those shirts in these 3! = 6 different ways:
      5 then 7 then 13 or,
      5 then 13 then 7 or,
      7 then 5 then 13 or,
      7 then 13 then 5 or,
      13 then 5 then 7 or,
      13 then 7 then 5

      (8 votes)

  • Barbara DiLucchio

    6 years agoPosted 6 years ago. Direct link to Barbara DiLucchio's post “when you are writing pyth...”

    when you are writing python code for any given factorial say one that a user enters like 11. what if you also need to determine which values are odd and which are even is it true that they are always even? do you ever get odd results going down thru the range values?

    (3 votes)

    • Jesse

      5 years agoPosted 5 years ago. Direct link to Jesse's post “All the results of the fa...”

      All the results of the factorial function are always even UNLESS it's 1! or 0!. Therefore for all n! such that n doesn't equal 0 or 1 the result is even. That's because 0! and 1! equals 1, which is odd.

      This is because an odd number multiplied by an even number is even, and an even number multiplied by an even number is also even. Therefore for every n! where n > 1 an even number is involved, making the answer even.

      Hope that helped!

      (3 votes)

  • Renan Teixeira Ferraz

    6 years agoPosted 6 years ago. Direct link to Renan Teixeira Ferraz's post “The autograder isn't work...”

    The autograder isn't working for me. The console is showing me that the code works, but the autograder doesn't.

    (3 votes)

    • Jesse

      5 years agoPosted 5 years ago. Direct link to Jesse's post “That happened to me too. ...”

      That happened to me too. I reported it and am hoping they'll fix the issue. (the console showed it worked, and my code is just like the code in the hint).

      (2 votes)

  • cmb717

    3 years agoPosted 3 years ago. Direct link to cmb717's post “I can not get the next ch...”

    I can not get the next challenge completed what am i doing wrong? Here is my code (edited to only show relevant parts)

    var result = 1;

    for(var i= result; i>=n; i--) {
    n=i*n;
    }


    I looked through the comments for help but all I could find were Cameron's cryptic answers can I have a more firm hint please. (not saying Cameron did a bad job just saying I can not understand their hints)

    (2 votes)

    • Cameron

      3 years agoPosted 3 years ago. Direct link to Cameron's post “Let's review some for loo...”

      Let's review some for loop basics:

      //This for loop will:
      //-begin the counter at start,
      //-do some stuff each loop
      //-then increment counter
      //until the counter exceeds last
      for( var counter = start; counter <= last; counter++){
      //this is the body where we do some stuff
      }

      A simple for loop that would print out the numbers from 1 to 10 would be:

      for( var i = 1; i <= 10; counter++){
      println(i);
      }

      Many find the BASIC version of this a bit easier to understand:

      FOR I=1 TO 10
      PRINT(I)
      NEXT I

      There are a couple of things that you almost never want to do inside the body of the for loop:
      1) change the value of the counter
      2) change the value of the number you are stepping the counter towards e.g. last in the first example

      Doing either of those usually leads to unpredictable behaviour.

      Here's some extra hints using the hint code:

      var factorial = function(n) {
      //this will set the value of result before we start looping
      var result = ;

      //think about calculating n! by hand
      // n! = 1 * 2 * 3 * ... * n-2 * n-1 * n
      // what number do we start at ?
      // what number do we end at ?
      for(var i = ; i <= ; ) {

      //we need to update the value of result here, so that it
      //will be equal to n! by the time we stop looping

      //changing the counter i, or the value that it is going to, would be bad here
      ;
      }

      //we are returning the value of result here
      //so result should now hold the value of n!
      return result;
      };

      Good Luck

      (3 votes)

  • Matthew Narodowg

    4 years agoPosted 4 years ago. Direct link to Matthew Narodowg's post “my code keeps coming up w...”

    my code keeps coming up with this
    message

    Hm, are you hard-coding the number to count up to?
    idk what i'm doing wrong

    (2 votes)

    • Cameron

      4 years agoPosted 4 years ago. Direct link to Cameron's post “If you put a number in th...”

      If you put a number in the condition of the for loop you'll get that. e.g. i <= 5

      If you put a number like 5 instead of using a variable, that is what is called "hard coding". Hard coding makes your program work for only one specific input. We want our programs to be flexible and work with different inputs, so we use variables.

      (2 votes)

  • SecretCoder

    2 years agoPosted 2 years ago. Direct link to SecretCoder's post “Why do they have a '!' af...”

    Why do they have a '!' after numbers like: 0!, 1!, n!

    (2 votes)

    • Cameron

      2 years agoPosted 2 years ago. Direct link to Cameron's post “From the first paragraph ...”

      From the first paragraph of the article:
      "We indicate the factorial of n by n!. It's just the product of the integers 1 through n. For example, 5! equals 1 * 2 * 3 * 4 * 5 or 120. (Note: Wherever we're talking about the factorial function, all exclamation points refer to the factorial function and are not for emphasis.)"

      (2 votes)

  • Eden Washington

    2 years agoPosted 2 years ago. Direct link to Eden Washington's post “I can't get this complete...”

    I can't get this completed.. I've done the coding statements as I knew how to do

    But I kept on getting
    "Hm, do all your assertions pass?" for this code I was doing, what do i need to add or fix?

    var result = 1;
    for(var i= 0; i<=n; i++) {
    result=(n+1)*(i+1);
    }
    return result;
    };

    (2 votes)

    • Cameron

      2 years agoPosted 2 years ago. Direct link to Cameron's post “The above calculation `re...”

      The above calculation result=(n+1)*(i+1); is incorrect.
      A few things to remember:
      n! = 1 * 2 * 3 * ... * n
      and
      n! = n * (n-1)!
      and
      0! = 1 (which is same as result before the for loop. So, if it skipped the calculations in the for loop it would be nice)

      (2 votes)

The factorial function (article) | Khan Academy (2024)

FAQs

The factorial function (article) | Khan Academy? ›

It's just the product of the integers 1 through n. For example, 5! equals 1 * 2 * 3 * 4 * 5 or 120. (Note: Wherever we're talking about the factorial function, all exclamation points refer to the factorial function and are not for emphasis.)"

What is the factorial function? ›

In short, a factorial is a function that multiplies a number by every number below it till 1. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3 × 2 × 1 and is equal to 6.

Why is 0 factorial 1 khan academy? ›

Factorials are a quick way to represent multiplying a number by all the smaller positive integers down to one. This video also shows why mathematicians have defined zero factorial as one, instead of zero, to make the formula for permutations work in all cases.

What are factorials Khan Academy? ›

Factorials represent the product of a series of descending numbers, marked with an exclamation point. For example, 4 factorial (4!) equals 24, because 4 times 3 times 2 times 1 equals 24.

Why do we use factorial in permutation? ›

A "permutation" uses factorials for solving situations in which not all of the possibilities will be selected.

How do you explain factorial? ›

We denote factorial with an exclamation point, and it simply tells us to multiply any natural number by all the natural numbers that are smaller than it. If we're asked to evaluate 5!, I simply have to do 5 * 4 * 3 * 2 * 1, and I get 120. 9! is 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 =362,880.

What is the purpose of the factorial? ›

We use factorials when we look at permutations and combinations. Permutations tell us how many different ways we can arrange things if their order matters. Combinations tells us how many ways we can choose k item from n items if their order does not matter.

What is factorial used in real life? ›

Factorials can be simple to compute and have many practical applications in the real world. For example, some companies use factorials to look at permutations and combinations for business purposes, like determining the number of trucks needed to supply their stores in each district.

How big is 52 factorial? ›

52! is approximately 8.0658e67. For an exact representation, view a factorial table or try a "new-school" calculator, one that understands long integers. A billion years currently equals 3.155692608e16 seconds; however, the addition of leap seconds due to the deceleration of Earth's orbit introduces some variation.

Is factorial of 0 possible? ›

Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. But there are no positive values less than zero so the data set cannot be arranged which counts as the possible combination of how data can be arranged (it cannot). Thus, 0! = 1.

What are factorials for dummies? ›

To find a number's factorial, you write all the whole numbers from 1 to that number and then multiply them all together. For example, 5! (read as "five factorial") is shorthand for 1 × 2 × 3 × 4 × 5, which you can work out on your calculator to get the value 120.

Why is zero factorial 1? ›

A factorial is the product of all integers less than or equal to the original number. As such, zero factorial equals one because it represents the one possible arrangement of an empty set: none at all. Factorials are used to determine permutations, which represent unique orders of elements in a set.

What grade math is factorial? ›

IXL | Factorials | 7th grade math.

Which grows faster, factorial or exponential? ›

Factorials will always overtake exponentials because they multiply by bigger and bigger numbers at each stage. Factorials help to count the number of ways to do something. For example, if you have five songs on a playlist, there are different sequences in which you could listen to them without repetition.

What is the advantage of factorial? ›

Factorial designs offer several advantages over other types of DOE, such as one-factor-at-a-time (OFAT) designs. These advantages include being more efficient and informative, as they allow for testing multiple factors and interactions in fewer experimental runs than OFAT designs.

What is the factorial rule? ›

The factorial function (symbol: !) says to multiply all whole numbers from our chosen number down to 1. Examples: 4! = 4 × 3 × 2 × 1 = 24.

What is factorial 5 function? ›

Factorial of a positive integer (number) is the sum of multiplication of all the integers smaller than that positive integer. For example, factorial of 5 is 5 * 4 * 3 * 2 * 1 which equals to 120.

What is a full factorial function? ›

A full factorial design is a simple systematic design style that allows for estimation of main effects and interactions. This design is very useful, but requires a large number of test points as the levels of a factor or the number of factors increase.

What is a factorial of 10? ›

10 factorial is nothing but 10 × 9 × 8 × 7 × 6 × 5 × 4 ×3 × 2 × 1 = 3,628,800.

Top Articles
Latest Posts
Recommended Articles
Article information

Author: Terrell Hackett

Last Updated:

Views: 5609

Rating: 4.1 / 5 (72 voted)

Reviews: 95% of readers found this page helpful

Author information

Name: Terrell Hackett

Birthday: 1992-03-17

Address: Suite 453 459 Gibson Squares, East Adriane, AK 71925-5692

Phone: +21811810803470

Job: Chief Representative

Hobby: Board games, Rock climbing, Ghost hunting, Origami, Kabaddi, Mushroom hunting, Gaming

Introduction: My name is Terrell Hackett, I am a gleaming, brainy, courageous, helpful, healthy, cooperative, graceful person who loves writing and wants to share my knowledge and understanding with you.