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Question
Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4, and 5 repetitions not allowed?
Solution
The given numbers are 1, 2, 3, 4, 5
The total number of arrangements.
Using the digits 1, 2, 3, 4 and 5 taking 4 at a time is 5P4
= 5 × 4 × 3 × 2
= 120
∴ 120 four-digit numbers can be formed using the given 5 digits without repetition.
To find the sum of these numbers.
We will find the sum of digits at unit’s, ten’s, hundred’s and thousand’s place in all these 120 numbers.
Consider the digit in unit’s place. In all these numbers
Each of these digits 1, 2, 3, 4, 5 occurs 120 in `120/5`
= 24 times in the units place
∴ The sum of the digits at unit’s place
= 24(1 + 2 + 3 + 4 + 5)
= 24 × 15
= 360
Similarly sum of the digit’s at ten’s place = 360
Sum of the digit’s at hundred’s place = 360
Sum of the digit’s at thousand’s place = 360
∴ Sum of all four digit numbers formed using the digit’s 1, 2, 3, 4, 5
= 360 × 10° + 360 × 101 + 360 × 102 + 360 × 103
= 360(10° + 101 + 102 + 103)
= 360(1 + 10 + 100 + 1000)
= 360 × 1111
= 3,99,960
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