The 10 numbers 0 through 9 are to be used in code groups of

The 10 numbers 0 through 9 are to be 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 That is, the same number cannot be used twice (or more) In a total sequence. For example, 2256, 2562, or 5559 would not be permitted. How many different code groups can be designed?

Solution

The problem is equivalent to filling up 4 blank spaces with numbers from 0 to 9 without repititions.

The first blank can be filled up using any number from 0 to 9 in 10 ways. When any number is placed in the first blank, the second blank can be filled by any number from 0 to 9 except for the number in first place in 9 ways. When fist two places are filled, the third space can similarly be occupied in 8 ways. And at last the 4th blank in 7 ways.

Hence there can be a total of 10*9*8*7 =5040 different code groups.