If 4-digit numbers greater than 5,000 are randomly formed from the digits 0, 1, 3, 5, and 7, what is the probability of forming a number divisible by 5 when, (i) the digits are repeated? (ii) the repetition of digits is not allowed?
The four digit numbers greater than 5000 are randomly formed from the digits 0 , 1 , 3 , 5 , 7 .
(i)
Given, the digits are repeated.
The number formed is greater than 5000 , the leftmost digit is either 7 or 5 . The remaining three places can be filled by any of the digits 0 , 1 , 3 , 5 or 7 .
Total four digit number formed greater than 5000 is 2 × 5 × 5 × 5 − 1 = 249 .
If unit digit of a number is 0 or 5 then number is divisible by 5.
Total numbers that are divisible by 5 is 2 × 5 × 5 × 2 − 1 = 99 .
Assume E be the event when number is divisible by 5 and digits are repeating.
The probability is,
P ( E ) = 99 249 = 33 249
Thus, the probability of number divisible by 5 when digits repeated is 33 249 .
(ii)
Given, repetition of digit is not allowed. Then the thousands place can be filled with digits either 5 or 7 and remaining three places can be filled with any of the remaining four digits.
Total four digit number formed greater than 5000 is 2 × 4 × 3 × 2 = 48 .
If the digit at thousand place is 5 then unit place can be filled with 0 and tens and hundreds places can be filled with any of the remaining three digits.
The four digits number formed started with 5 and divisible with 5 is 3 × 2 = 6 .
If the digit at thousand place is 7 then unit place can be filled with 0 or 5 .
The four digit numbers starting with 7 and divisible by 5 is 1 × 2 × 3 × 2 = 12 .
Total number greater than 5000 and divisible by 5 is,
6 + 12 = 18
The probability of a number formed is divisible by 5 with repetition of digits is,
18 48 = 3 8
Thus, the required probability is 3 8 .
The four digit numbers greater than
(i)
Given, the digits are repeated.
The number formed is greater than
Total four digit number formed greater than
If unit digit of a number is 0 or 5 then number is divisible by 5.
Total numbers that are divisible by
Assume E be the event when number is divisible by
The probability is,
Thus, the probability of number divisible by
(ii)
Given, repetition of digit is not allowed. Then the thousands place can be filled with digits either
Total four digit number formed greater than
If the digit at thousand place is
The four digits number formed started with
If the digit at thousand place is 7 then unit place can be filled with
The four digit numbers starting with
Total number greater than
The probability of a number formed is divisible by 5 with repetition of digits is,
Thus, the required probability is
