What will be the sum of adding all possible four-digit number formed by 7,9,1,3 using each of the digit only once?
a-133220 b-122320 c-133230 d-133320
Answers
Answer:
(d) 133320
Step-by-step explanation:
Using the given 4 numbers, we can make 4! = 24 four-digit numbers without digit repetition.
I'll write that 24 numbers.
7913, 7931, 7139, 7193, 7391, 7319,
9137, 9173, 9371, 9317, 9713, 9731,
1379, 1397, 1793, 1739, 1937, 1973,
3791, 3719, 3917, 3971, 3179, 3197
Among these 24 numbers, there are 4-digit numbers each in which each numbers 7, 9, 1, 3 are repeated 6 times at its ones place.
Given are the numbers in which 7 is repeated 6 at ones place.
9137, 9317, 1397, 1937, 3917, 3197
When we add the numbers at ones place of these numbers, we get 6 × 7 = 42.
Also there are 6 numbers each in which 9, 1, 3 are repeated at ones place.
For numbers in which 9 is repeated, the sum of numbers at ones place is 6 × 9 = 54.
For numbers in which 1 is repeated, the sum of numbers at ones place is 6 × 1 = 6.
For numbers in which 3 is repeated, the sum of numbers at ones place is 6 × 3 = 18.
Add these all.
42 + 54 + 6 + 18 = 6(7 + 9 + 1 + 3) = 6 × 20 = 120
This 120 is got by adding the digits at ones place of the 24 four-digit numbers. 0 is taken as the ones digit of the sum asked and 12 will be carried as remainder.
When we consider tens digit, the same feature of ones digit is shown here. That is, there are 6 numbers each in which 7, 9, 1, 3 repeats at tens place, and by adding these tens digits, we get 120 too.
The same is occurring in the hundreds and thousands places too.
So I'm writing the sum decimally as shown below.
[Writing decimally means to write by separating the digits by a slash (/) or anything else. Here, I'm using / to separate the digits. Remember that the digits are not divided.]
120 / 120 / 120 / 120
Carry 12 towards as remainder and consider 0 as the corresponding digits.
120 + 12 / 0 + 12 / 0 + 12 / 0
132/12/12/0
From tens and hundreds digits 12 each, carry 1 as remainder.
132 + 1 / 2 + 1 / 2 / 0
133 / 3 / 2 / 0
Remove the slashes and we get,
133320
So option (d) is the right answer.
Okay, let me prefer you a simple method to find it.
TO REMEMBER...
The sum of all possible 4-digit numbers formed by a, b, c, d without repetition is,
6666(a + b + c + d)
In this question,
a = 7 ; b = 9 ; c = 1 ; d = 3 6666(a + b + c + d) 6666(7 + 9 + 1 + 3)
6666 × 20
133320
Hope this article may be helpful.
If you've any doubts, please don't rate it 1 star, 2 star, etc. It'll make me cry, let this answer go in vain and lose my precious time. So you may ask me in the 'comments' section and I'll be able to clear it.
Thank you. Have a nice day. :-)
#adithyasajeevan