Question:

The sum of all $4$ digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

Updated On: Jun 23, 2023

$ 133320 $
$ 123330 $
$ 113230 $
$ 323430 $

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The Correct Option is A

Solution and Explanation

There are $4! = 24$ numbers. Each digit occurring $3! = 6$ times, in the unit's, ten's, hundred?? and thousand's places. We note that $6(2 + 4 + 6 + 8) = 120$. Thus in the over all sum there will be $120$ units, $120$ tens, $120$ hundreds and $120$ thousands. $\therefore$ The required sum $= 120 (1 + 10 + 10^2 + 10^3)$ $ = 120 \times 1111 = 133320$.

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Concepts Used:

Arithmetic Progression

Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.

For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.

In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.

For eg:- 4,6,8,10,12,14,16

We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.

Read More: Sum of First N Terms of an AP