How Many Four-Digit Odd Numbers Have Digits Only From the Set 1, …, 5?

Probability is a statistical term that tells what chances for an event to occur are. We have several counting methods for evaluating the Probabilities like Permutation, combination, factorization etc. Calculation of Probability depends completely on type of events we are referring to. Events can be mutually exclusive (independent events) or mutually dependent events. We may need to remember all these basics while solving such problems. To know we have two possibilities: how many four digit odd Numbers have digits only from the Set 1 to 5.
1.       Repetition is allowed and
 
2.       Repetition is not allowed
Let us take first case: For a 4 digit number to be odd, we need to have digits 1, 3 and 5 in first place (units place) from the set (1, 2, 3, 4, 5). The number of ways these three digits can be chosen is 3.
Next we come to rest of three places in the number. As we have already considered that repetition is allowed, any of 5 digits can be used to fill these places. So number of ways we can make an odd 4 digit number is: 5 * 5 * 5 * 3 = 375.
Second case says that no repetition is allowed: In this case also the next place digit of a 4 digit number can be filled in 3 possible ways using the digits 1, 3 and 5. After this we are left with 4 digits in the given set. At first considering the tens place, we can fill it in 4 possible ways. Next comes the hundreds place which can be filled in 3 possible ways and last is thousands place which can be filled in 2 possible ways. So, total number of ways we can make an odd 4 digit number when no repetition is allowed is: 2 * 3 * 4 * 3 = 72 ways.