How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed? (ii) repetition of the digits is not allowed?
There are many ways to form 3-digit numbers from the digits 1, 2, 3, 4 and 5.
(i)
The unit’s place can be filled by any of the five digits. Since, the repetition of the digits is allowed, thus, the remaining two places that are ten’s and hundred’s places can also be filled by any of the five digits.
The number of ways in which the three-digit numbers can be formed from the given digits can be calculated by multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, then the total number of occurrence of the events in the given order is m × n . Since, each place have 5 ways to obtain any of the given digits.
The total number of ways to form three digit numbers with repetition is,
5 × 5 × 5 = 125
Thus, 3-digit number can be formed in 125 ways.
(ii)
If unit’s place is filled first, then it can be filled by any of the five digits. Thus, the number of ways of filling the unit’s place of the three-digit number is 5. Since the repetition is not allowed in this case so, there remains only 4 digits from the given digits because any one from the given digits has occupied unit’s place.
Now, the ten’s place can be filled by any of the remaining 4 digits. The possible ways of filling the ten’s place is 4. Then for the hundred’s place, there are only 3 digits left. Thus, the number of ways of filling the hundred’s place is 3.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, then the total number of occurrence of the events in the given order is m × n .
The number of ways in which the three-digit numbers can be formed from the given digits is,
5 × 4 × 3 = 60
Thus, 3-digit numbers can be formed in 60 ways without repetition.
There are many ways to form 3-digit numbers from the digits 1, 2, 3, 4 and 5.
(i)
The unit’s place can be filled by any of the five digits. Since, the repetition of the digits is allowed, thus, the remaining two places that are ten’s and hundred’s places can also be filled by any of the five digits.
The number of ways in which the three-digit numbers can be formed from the given digits can be calculated by multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, then the total number of occurrence of the events in the given order is
The total number of ways to form three digit numbers with repetition is,
Thus, 3-digit number can be formed in 125 ways.
(ii)
If unit’s place is filled first, then it can be filled by any of the five digits. Thus, the number of ways of filling the unit’s place of the three-digit number is 5. Since the repetition is not allowed in this case so, there remains only 4 digits from the given digits because any one from the given digits has occupied unit’s place.
Now, the ten’s place can be filled by any of the remaining 4 digits. The possible ways of filling the ten’s place is 4. Then for the hundred’s place, there are only 3 digits left. Thus, the number of ways of filling the hundred’s place is 3.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, then the total number of occurrence of the events in the given order is
The number of ways in which the three-digit numbers can be formed from the given digits is,
Thus, 3-digit numbers can be formed in 60 ways without repetition.
