Question 1087849: 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 by mathmate(429)   (Show Source):
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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?
Solution:
Since the digits represent a code, leading zeroes are permitted, such as 0328.
However, repetition is not allowed in the code, so
first digit has 10 choices [0-9]
second digit has 9 choices [0-9, less the first digit]
third digit has 8 choices [0-9, less the first & second digits]
fourth digit has 7 choices [0-9, less the first three digits]
The total number, N, of codes is therefore, by the multiplication rule,
N=10*9*8*7=P(10,4)=5040 codes.
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