How many four digit numbers can be formed using every digit only once in the number?
TLDR - 5040 numbers can be formed using every digit (0-9) only once.
I had a very interesting discussion with @awesomianist regarding this question. It feels good to revise our high school Maths.*
This question can be easily answered if we understand Permutation and Combination. Permutation is just an ordered Combination. In Permutation, the order matters, whereas in Combination, the order does't matter.
So there are 2 types of Permutation. One is where repetition is allowed, and another one is where repetition is not allowed.
To answer this question, we will have to use Permutation since the order matters (1234 and 4321 is 2 different numbers) and repetition is not allowed (1123 uses 1 twice).
The formula for this would be: n!/(n − r)!
But my way of understanding permutation is this:
I imagine 4 empty chairs representing the 4 digit numbers. Now I name them seat A, B, C and D.
_ _ _ _
And there are 10 people, each representing the digit 0 - 9.
10 people can choose to sit on seat A, but once a person sit on seat A, the next 9 people can choose to sit on seat B. Now there's 8 people left and they can choose to sit on seat C and the 7 people that are left can now choose to sit on seat D.
Now we only need to multiply 10 by 9 by 8 by 7 (10*9*8*7), which equals to 5040.
So there are 5040 numbers can be formed using every digit only once in the number.
It was fun going through the whole thought process and spending a whole A4 paper worth of scribbles to come to that conclusion, only to realize it's taught in O-level high school 😂😂
You were right the first time when you said (10x9x8x7=5040) solves the question but i insist on finding an explaining for it.. hahahahaha
In case people don't understand (n!/(n-r)!)
n = the number of total "digits" to choose from. Since digits are 0,1,2,3,4,5,6,7,8 and 9, there are 10 of them. n=10r = how many digit combination, so from the question, OP asks 4 digit letter. r=4
applying the numbers into the formula, you get (10!/(10-4)!). So rounds out to 10!/6!.
in case you dont know, "!" is a factorial, so when i say "10!" it means 10x9x8x7x6x5x4x3x2x1.
plug that into your calculator (or do it yourself), you will find that 10!/6!=5040, is basically 10x9x8x7= 5040Just shows that we actually listen to our Maths teacher in class! :D
Hahaha... First time I read about math again after 2 years 😆.
Thanks for reminding me to refresh my brain.
Same here actually :D