Given:
Digits available: 0, 1, 3, 5, 7
Four–digit numbers greater than 5000 are formed.
A number is divisible by 5 if its last digit is 0 or 5.
Total possible cases
Since the number must be greater than 5000, the thousand’s digit can be:
5 or 7
(i) Digits may be repeated
Total number of outcomes:
Thousand’s digit → 2 choices (5, 7)
Hundreds digit → 5 choices
Tens digit → 5 choices
Units digit → 5 choices
Total outcomes = 2 × 5 × 5 × 5 = 250
Favourable outcomes (divisible by 5):
Units digit must be 0 or 5 → 2 choices
Thousand’s digit → 2 choices
Hundreds digit → 5 choices
Tens digit → 5 choices
Favourable outcomes = 2 × 5 × 5 × 2 = 100
Probability:
P = 100 / 250 = 2 / 5
(ii) Repetition of digits is not allowed
Total number of outcomes:
Thousand’s digit → 2 choices (5, 7)
Remaining digits for hundreds, tens, units → 4, 3, 2 choices respectively
Total outcomes = 2 × 4 × 3 × 2 = 48
Favourable outcomes (divisible by 5):
Case 1: Units digit = 0
Thousand’s digit → 2 choices
Remaining digits → choose 2 from remaining 3 digits in order
Ways = 2 × 3 × 2 = 12
Case 2: Units digit = 5
Then thousand’s digit must be 7 (since repetition is not allowed)
Remaining digits → choose 2 from remaining 3 digits in order
Ways = 1 × 3 × 2 = 6
Total favourable outcomes = 12 + 6 = 18
Probability:
P = 18 / 48 = 3 / 8
Final Answers:
(i) Probability = 2 / 5
(ii) Probability = 3 / 8
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤5.
State true or false: (give reason for your answer)
(i) A and B are mutually exclusive
(ii)A and B are mutually exclusive and exhaustive
(iii) A=B′
(iv)A and C are mutually exclusive
(v)A and B′ are mutually exclusive.
(vi) A′,B′,C are mutually exclusive and exhaustive.
From the employees of a company, \(5\) persons are selected to represent them in the managing committee of the company. Particulars of five persons are as follows: A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over \(35\) years?