A four-digit number is to be formed using the digits 0, 1, 2, 3, 4, 5. All the
digits are to be different. Find the probability that the digit formed is
(i) odd, (ii) greater than 4000, (iii) greater than 3400, and (iv) a multiple
of 5.
Answers
Given : A four-digit number is to be formed using the digits 0, 1, 2, 3, 4, 5
All the digits are to be different
To Find : the probability that the digit formed is
(i) odd, (ii) greater than 4000,
(iii) greater than 3400, and (iv) a multiple of 5
Solution:
digits 0, 1, 2, 3, 4, 5
case 1 : 1 of the digits is 0
rest 3 Digits can be selected in ⁵C₃ = 10 ways
Numbers can be be formed in 3 * 3! = 18 ways
= 18 * 10
= 180
case 2 : 0 is not the digit
rest 4 Digits can be selected in ⁵C₄ = 5 ways
Numbers can be be formed in 4! = 24 ways
= 24 * 5
= 120
Total Possible numbers = 180 + 120 = 300
odd
Ending with 1 , 3 , 5 ( 3 ways) Having 0 rest 2 number in ⁴C₂ = 6 ways
2 * 2! = 6 ways hence 3 * 6 * 6 = 108
Ending with 1 , 3 , 5 ( 3 ways) not having 0 rest 3 number in ⁴C₃ = 4 ways
3! = 6 ways hence 3 * 4 * 6 = 72
= 108 + 72 = 180
Probability = 180/300 = 3/5
greater than 4000
1st Digit 4 or 5 2 ways Hence 2 * ⁵C₃ * 3! = 120 ways
120/300 = 2/5
greater than 3400
Having 1st digit 4 or 5 Hence 2 * ⁵C₃ * 3! = 120 ways
1st digit 3 , 2nd digit 4 or 5 1 * 2 * ⁴C₂ * 2! = 24 ways
Probability = 144/300 = 48/100 = 12/25
a multiple of 5
last Digit 0 then ⁵C₃ * 3! = 60
last Digit 5
then if 0 is one of the digits then ⁴C₂ * 2 * 2! = 24 Ways
if 0 is not one of the digit then ⁴C₃ * 3! = 24 ways
= 60 + 24 + 24 = 108
Probability = 108/300 = 9/25
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