Consider UPC codes on products which have 11 digits of the form NXXXX-NXXXXX, wh

Question

Consider UPC codes on products which have 11 digits of the form NXXXX-NXXXXX, where the first 5 digits NXXXX are the company code and the last 6 digits NXXXXX are the product code. Here, N represents a digit from 2 to 6, inclusive, and X represents any digit 0 to 9. For any part of this question, you may leave your answer with operators in it (e.g., 5 * 10 + 9/3 would be acceptable or you may work it out to get 53). For one fixed company code, how many different product codes can we form (these have the form NXXXXX)? Using your answer for the previous part, how many total UPCs can we construct (using all combinations of company and product codes)? For one fixed company code, how many product codes can we form where each digit is unique? For one fixed company code, how many product codes can we form with no consecutive digits the same? (e.g., 255223 has 5 appear consecutively, as well as 2) Suppose we want to sell 25 million different products. What is the fewest number of company codes needed to guarantee that each product can be assigned a unique UPC code?

Explanation / Answer

a)

Product is 6 digit code. N is from 2 to 6 so we can possible use any of the 5 numbers. X is from 0 to 9 so we can use possible 10 numbers. These 10 numbers can occupy 5 places and 5 numbers can occupy first place only so answer is

5*10*10*10*10*10 = 5*105

b)

Company code is made of 5 digits, where first digit is N and rest of 4 is X. So first place can be filled by 5 digits and second can be again 10 digits so total number of possible company code is

5*104

and we had product code = 5*105

so total number of possible UPC code is company code * product code

25*109.

c)

For this we have to find all combination where each digit should be unique. As first place can have max of 5 number from 2 to 6 and rest of 5 places can have 0 to 9 so 10 numbers

so N can have 2 to 6 so 5 number but first X will have one less so that it will not match N. So suppose N took 2 so first X can have 0,1,3,4,5,6,7,8,9 and when n took 3 it will have all the numbers except 3 similarly 2nd X will have all the numbers except the number occupy by first X and N .

So from that we can say we have total number of code = 5*9*8*7*6*5 = 75600

d) We have to care about non-consecutive digits. So if N occupy 2 then its adjacent X should have all numbers except 2 then its adjacent number could have all numbers from 0 to 9 except which X has but remember it can have value of N. So each adjacent elements can't have same digit so only one digit is less than problem a. So answer is

5*9*9*9*9*9 = 295245

e)

From calculation of problem 'a' we know that each company code have .5millions product code. So 10 company code will help in 5 milions unique UPC code. And so 50 will help in 25 million product code. So just put 2 to 6 in N and put 0 to 9 in last X and fill rest of X with 0

So 20000 20001 20002.....20009 and 30001 30002 30003... 30009...... 60001 60002..... 60009 so there 50 company codes and 5*105 product code. So answer is 50*5*105 = 25*106

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