Answer: (a) 0.28
(b) 0.89
(c) 0.22
Explanation:
Since repetition is not allowed, and we are forming 4 digits number from 1,2 ,…9
The first digit can be any of the 9 digits
The second digit can be any of the remaining 8 digits
The third digit can be any of the remaining 7 digits
The forth digit can be any of the remaining 6 digits
Therefore arrangement of the 4-digit number implies
Arrangement = 9 x 8 x 7 x 6
= 3024
It can also be written as: 9P4, which is also 3024.
The next thing is to get 4 –digit number ≥3500 that can be formed from the digits 1 , 2 …9
For the 4 – digits to be ≥ 3500, then the first digit will be either of the following set { 3,4,5,6,7,8,9)
The first digit can be any of the 7 digits
The second digit can be any of the remaining 6 digits
The third digit can be any of the remaining 5 digits
The forth can be any of the remaining 4 digits
Therefore, arrangement = 7 x 6 x 5 x 4 = 840
(a) p( number ≥3500) = number≥3500/3024
= 840/3024
= 0.28
(b) If the first digit must be even, the it must be one of the following set { 2,4,6,8}
Once the first digit has been filled , we have eight digits remaining
Second digit can be any of these eight digits
Third digit can be any of the remaining seven digits
Forth digit can be any of the remaining six digits
Therefore , Arrangement = 4 x 8 x 7 x 6 = 1344
P( number begin with even digit) = 1344/3024
Also, If the last digit must be even, then it must be one of the following set { 2,4,6,8}
Once the last digit has been filled , we have eight digits remaining
First digit can be any of these eight digits
Second digit can be any of the remaining seven digits
Third digit can be any of the remaining six digits
Therefore , Arrangement = 4 x 8 x 7 x 6 = 1344
P( number ends with even digit) = 1344/3024
P(number begins or end with even digits ) = p( number begin with even digit) + p( number ends with even digit) = 1344/3024 + 1344/3024
2688/3024
= 0.89
(c) To get exactly two even number
Since there are four even numbers, it could be the combination of any 2
Therefore , number of exactly 2 even numbers = 2 x 8 x 7 x 6 = 672
Therefore p( exactly 2 even number ) = 672/3024
= 0.2