A positive integer less than 100000 is picked at random a Wh

A positive integer less than 100,000 is picked at random.

(a) What is the probability that the integer contains the digit 3 at least once?

(b) What is the probability that the integer contains 5 different non-zero digits?

Ans(a):

A positive integer less than 100,000 means total possible numbers are 99999

now we count how many favorable cases ( contains the digit 3 at least once)

for 1 digit number:

3 is the only number so no of digits =1

for 2 digit number:

let\'s find number that dont contain any digit as 3

first digit can be filled in 8 ways using (1,2,4,5,6,7,8,9) so 8 ways

second digit can be filled in 9 ways using (0,1,2,4,5,6,7,8,9) so 9 ways

so total number of arrangements =8*9=72

Hence numbers that contain at least one 3= 9*10-72=18

for 3 digit number:

let\'s find number that don\'t contain any digit as 3

first digit can be filled in 8 ways using (1,2,4,5,6,7,8,9) so 8 ways

second digit can be filled in 9 ways using (0,1,2,4,5,6,7,8,9) so 9 ways

third digit can be filled in 9 ways using (0,1,2,4,5,6,7,8,9) so 9 ways

so total number of arrangements =8*9*9

Hence numbers that contain at least one 3= 9*10*10-8*9*9=252

for 4 digit number:

similarly numbers that contain at least one 3= 9*10*10*10-8*9*9*9=3168

for 5 digit number:

similarly numbers that contain at least one 3= 9*10*10*10*10-8*9*9*9*9=37512

Hence total favorable cases are =37512+3168+252+18+1=40951

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Hence probability that the integer contains the digit 3 at least once=40951/99999

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Ans(b):

contains 5 different non-zero digits means picked number is 5 digit number

first digit can be filled in 9 ways using (1,2,3,4,5,6,7,8,9)

second digit can be filled by remaining 9 digits

third digit can be filled by remaining 8 digits

fourth digit can be filled by remaining 7 digits

fifth digit can be filled by remaining 6 digits

so favorable number of cases are =9*9*8*7*6=27216

Then probability that the integer contains 5 different non-zero digits = 27216/99999

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Ans(c):

There are total 316 perfect squares less than 100000

then probability that the integer is a perfect square=316/99999