How many combinations with 4 numbers 1-24
Example: 4! is shorthand for 4 × 3 × 2 × 1 Show
We usually say (for example) 4! as "4 factorial", but some people say "4 shriek" or "4 bang" Calculating From the Previous ValueWe can easily calculate a factorial from the previous one: As a table:
Example: 9! equals 362,880. Try to calculate 10!10! = 10 × 9! 10! = 10 × 362,880 = 3,628,800 So the rule is: n! = n × (n−1)! Which says "the factorial of any number is that number times the factorial of (that number minus 1)" So 10! = 10 × 9!, ... and 125! = 125 × 124!, etc. What About "0!"Zero Factorial is interesting ... it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together results in 1, but let's follow the pattern backwards from, say, 4! like this: And in many equations using 0! = 1 just makes sense. Example: how many ways can we arrange letters (without repeating)?
The formula is simply n! Now ... how many ways can we arrange no letters? Just one way, an empty space: So 0! = 1 Where is Factorial Used?One area they are used is in Combinations and Permutations. We had an example above, and here is a slightly different example: Example: How many different ways can 7 people come 1st, 2nd and 3rd?The list is quite long, if the 7 people are called a,b,c,d,e,f and g then the list includes: abc, abd, abe, abf, abg, acb, acd, ace, acf, ... etc. The formula is 7!(7−3)! = 7!4! Let us write the multiplies out in full: 7 × 6 × 5 × 4 × 3 × 2 × 14 × 3 × 2 × 1 = 7 × 6 × 5 That was neat. The 4 × 3 × 2 × 1 "cancelled out", leaving only 7 × 6 × 5. And: 7 × 6 × 5 = 210 So there are 210 different ways that 7 people could come 1st, 2nd and 3rd. Done! Example: What is 100! / 98!Using our knowledge from the previous example we can jump straight to this: 100!98! = 100 × 99 = 9900 A Small List
As you can see, it gets big quickly. If you need more, try the Full Precision Calculator. Interesting FactsSix weeks is exactly 10! seconds (=3,628,800) Here is why:
There are 52! ways to shuffle a deck of cards. That is 8.0658175... × 1067 Just shuffle a deck of cards and it is likely that you are the first person ever with that particular order. There are about 60! atoms in the observable Universe. 60! is about 8.320987... × 1081 and the current estimates are between 1078 to 1082 atoms in the observable Universe. 70! is approximately 1.197857... x 10100, which is just larger than a Googol (the digit 1 followed by one hundred zeros). 100! is approximately 9.3326215443944152681699238856 x 10157 200! is approximately 7.8865786736479050355236321393 x 10374 Advanced TopicsA Close Formula!n! ≈ (ne)n √2πn The "≈" means "approximately equal to". Let us see how good it is:
If you don't need perfect accuracy this may be useful. Note: it is called "Stirling's approximation" and is based on a simplifed version of the Gamma Function. What About Negatives?Can we have factorials for negative numbers? Yes ... but not for negative integers. Negative integer factorials (like -1!, -2!, etc) are undefined. Let's start with 3! = 3 × 2 × 1 = 6 and go down:
And from here on down all integer factorials are undefined. What About Decimals?Can we have factorials for numbers like 0.5 or −3.217? Yes we can! But we need to use the Gamma Function (advanced topic). Factorials can also be negative (except for negative integers). Half FactorialBut I can tell you the factorial of half (½) is half of the square root of pi . Here are some "half-integer" factorials:
It still follows the rule that "the factorial of any number is that number times the factorial of (1 smaller than that number)", because (3/2)! = (3/2) × (1/2)! Can you figure out what (7/2)! is? Double Factorial!!A double factorial is like a normal factorial but we skip every second number:
Notice how we multiply all even, or all odd, numbers. Note: if we want to apply factorial twice we write (n!)! 2229, 2230, 7006, 2231, 7007, 9080, 9081, 9082, 9083, 9084 What are all the combinations for 1234?1234, 1243, 1423, 4123, 1324, 1342, 1432, 4132, 3124, 3142, 3412, 4312, 2134, 2143, 2413, 4213, 2314, 2341, 2431, 4231, 3214, 3241, 3421, 4321.
How many combinations can I get with 4 numbers?There are 10,000 possible combinations that the digits 0-9 can be arranged into to form a four-digit code.
How many permutations does 1234 have?The answer is: There are 24 permutations. The 12 even permutations are: id , (1 2 3 4) , (1 3 2 4) , (1 4 2 3) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3).
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