Question

From the digits 0,1,2,3,5,7,8, and 9 , selected of them at random and from a four-digit number, none of these digits appears more than once in each number.

(a) Find the probability that the number formed is even.

(b) Find the probability that the number formed is divisible by 3.

Answer 1

Total number of possible four digit number from the given digits = 7*7*6*5 = 1470

(The thousands place can be filled with any of the given digits except 0, the hundreds place can be filled with any of the remaining 7 digits, tens place with any of the remaining 6 digits and units place with any of the remaining 5 digits)

Now, number of 4 digit even numbers that can be formed with these digits

So unit's place can have 0,2 or 8 for the resulting number to be even

Case 1: The unit's place contains 0

In this case total 4 digit even numbers = 7*6*5*1 = 210

Case 2: The unit's place contains 2 or 8

In this case total 4 digit even numbers = 6*6*5*2 = 360

Thus, total 4 digit numbers = 210 + 360 = 570

(a) The required probability = 570/1470 = 19/49

(b)

The sum of all the 4 digits should be divisible by 3 for the resulting number to be divisible by 3

Possible sums divisible by 3 from the given digits can be 6,9,12,15,18,21,24,27

For sum = 6, the digits used are 0,1,2,3, total such numbers = 3*3*2*1 = 18

For sum = 9, the digits used are 0,1,3,5, total such numbers = 18

For sum = 12, the digits used are 0,1,2,9 or 0,1,3,8 or 0,2,3,7, total such numbers = 18+18+18 = 54

For sum = 15, the digits used are 1,2,3,9 or 0,2,5,8 or 0,3,5,7 or 0,1,5,9 or 1,2,5,7 or 0,1,5,9, total such numbers = 24+18+18+18+24+18 = 120

For sum = 18, the digits used are 0,1,8,9 or 0,2,7,9 or 0,3,7,8 or 1,2,7,8 or 2,3,5,8 or 1,3,5,9, total such numbers =

18 + 18 + 18 + 24 + 24 + 24 = 126

For sum = 21, the digits used are 1,3,8,9 or 1,5,7,8 or 0,5,7,9 or 2,3,7,9 or 1,5,7,8, total such numbers = 24+24+18+24+24 = 114

For sum = 24, the digits used are 0,7,8,9 or 3,5,7,9 or 2,5,8,9, total such numbers = 18 + 24 + 24 = 66

For sum = 27, the digits used are 3,7,8,9 , total such numbers = 24

Thus, total 4 digit numbers divisible by 3 = 18 + 18 + 54 + 120 + 126 + 114 + 66 + 24 = 540

Thus, the required probability that the number formed is divisible by 3 = 540/1470 = 18/49