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

Step-by-step 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 :-

$^nP_r=\dfrac{n!}{(n-r)!}$

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

$^{10}P_{4}=\dfrac{10!}{(10-4)!}\\\\=\dfrac{10\times9\times8\times7\times6!}{6!}\\\\=\dfrac{10\times9\times8\times7}{1}\\\\=5040$

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