A grocery store rewards card has a 7 digit number to identify the user. The firs

Question

A grocery store rewards card has a 7 digit number to identify the user. The first digit must be 1 or 2. remaining six digits take values randomly between 0 9 inclusively. 6. The a) b) c) d) e) How many different ID numbers are possible for this rewards card? What is the probability that first three digits are all even (assume 0 is even)? What is the probability that the first three digits contain at least 1 even number? What is the probability that the ID number has no 3's or no 8's? What is the probability that all of the first four digits are distinct (i.e. no digit repeated)? There are no restrictions on the remaining digits.

Explanation / Answer

7 digit card number

first digit = 1 or 2

then first place be filled with only two choices either 1 or 2

but remaining six place can be filled with any of 10 digits each

total number of different sequence possible = 2*10^6=2000000

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b)

first digit will be even when only 2 is placed ,there is 1 choice

second and third digit have 5 choices each(0,2,4,6,8)

remaining 4 digits have 10 choices each(if there is no restriction on remaining digits and if there is restriction that other digits must be odd,then other 4 digits will have 5 choices)

so, total sequence with first three digits even = 1* 5 * 5 * 10^4=250000

so,P(first three digits even)=250000/2000000

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c) first three digits have no even number ,

then

first digit have choice that is 1

second and third digits have 5 choices each

other 4 digits can be even or odd so, total 10 choices each

so, total sequence with first three digit no even number = 1*5*5*10^4 = 250000

so, total sequence with first three digits have atleast 1 even number = 2000000 - 250000

P(first three digits have atleast 1 even number) = 1750000/2000000

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D)

no 3's,then 9 digits remaining

sequence with no 3's = 2*9^6

sequence with no 8's = 2*9^6

sequence with no 3's and no 8's = 2*8^6

P(A or B)=P(A)+P(B) - PA and B)

so,total sequence=  2*9^6 + 2*9^6 - 2*8^6 =1062882+1062882 - 524288

P( no 3's or no 8's) = 1601476/2000000

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for any doubts,ask in comment section..

please note that as per chegg answering guidlines,only four subparts can be answered

=0.125

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