Question 1 :
How many three-digit numbers are there with 3 in the unit place? (i) with repetition (ii) without repetition.
Solution :
The following numbers are used to form any number.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
By using the above numbers, we have to form a three digit number. In which unit place must contain the number "3".
(i) with repetition
Unit place ----> must be 3 ---> 1 option
Ten's place :
Including 3, we have 10 options
Hundred's place :
It should not contain 0. So, we have 9 options
Total numbers can be formed = 9 ⋅ 10 ⋅ 1 = 90
(ii) without repetition.
_____ _____ _____
Unit place ----> must be 3 ---> 1 option
Hundred's place :
By excluding 3 and 0, we have 8 options.
Ten's place :
Excluding 3 and the number filled in the hundreds place, we have 8 options.
Total numbers can be formed = 8 ⋅ 8 ⋅ 1 = 64
Question 2 :
How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5 ? if (i) repetition of digits allowed (ii) the repetition of digits is not allowed.
Solution :
We must form a three digit number.
(i) repetition of digits allowed
Unit place --> It may contain any of 6 numbers --> we have 6 options
Tens place --> It may contain any of 6 numbers --> we have 6 options
Hundreds place --> excluding 0 and 5 --> we have 4 options
Note : Any three digit number starts with 0. It should not 5, because we have to form a three digit number less than 500.
Total numbers can be formed = 4 ⋅ 6 ⋅ 6 = 144
(ii) the repetition of digits is not allowed.
Hundreds place :
Excluding 0 and 5 --> we have 4 options
Tens place :
Since repetition in not allowed including 0, we have 5 options.
Unit place :
Excluding the number filled in the tens and hundreds place, we have 4 options.
Total numbers can be formed = 4 ⋅ 5 ⋅ 4 = 80
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