Question

The numbers 0 through 9 are used in code groups of four to identify an item of clothing. Code 1083 might identify a blue blouse, size medium. The code group 2031 might identify a pair of pants, size 18, and so on. Repetitions of numbers are not permitted, i.e., the same number cannot be used more than once in a total sequence. As examples, 2256, 2562 or 5559 would not be permitted. How many different code groups can be designed?

Answer 1

Answer: 5040

Explanation:

Given : Number of choices to make a four digit code ( 0 to 9)= 10

If repetition is not allowed , then we use Permutations to find the number of different code groups can be designed.

The number of permutations of n things , taking r at a time is given by :-

`^nP_r=(n!)/((n-r)!)`

Then, the number of permutations of 10 numbers , taking 4 at a time is given by :-

`^(10)P_(4)=(10!)/((10-4)!)\\\\=(10*9*8*7*6!)/(6!)\\\\=(10*9*8*7)/(1)\\\\=5040`

Hence, the number of different code groups can be designed= 5040