Name: Application of Combinations
Uploaded: 2019-07-19
Duration: 7 min 48 s
Description: A combination is a technique that determines the number of possible arrangements in a collection of items where the order of selection does not matter. Let's...

Question

Find the sum of all

4

digit numbers that can be formed using the digits

1, 3, 5, 7, 9

.

Accepted Answer

We know that the number of $4$ digit numbers that can be formed using the digits $1, 3, 5, 7, 9$ is ${}^{5}P_{4} = 120$ .

We have to find their sum. We first find the sum of the digits in the units place of all the $120$ numbers. Put $1$ in the units place.

$1$

Similarly, each of the other digits $3, 5, 7, 9$ appears in the units place in ${}^{4}P_{3}$ times. The remaining $3$ places can be filled with the remaining $4$ digits in ${}^{4}P_{3}$ ways which means that there are ${}^{4}P_{3}$ number of $4$ digit numbers with $1$ in the units place.

Hence, by adding all these digits of the units place, we get the sum of the digits in the units place.

${}^{4}P_{3} \times 1 + {}^{4}P_{3} \times 3 + {}^{4}P_{3} \times 5 + {}^{4}P_{3} \times 7 + {}^{4}P_{3} \times 9$

$= {}^{4}P_{3} (1 + 3 + 5 + 7 + 9)$

$= {}^{4}P_{3} (25)$

Similarly, we get the sum of all digits in $10' s$ place also as ${}^{4}P_{3} \times 25$

Since it is in $10' s$ place, its value is

${}^{4}P_{3} \times 25 \times 10$

Like this the values of the sum of the digits in $100' s$ place and $1000' s$ place are respectively

${}^{4}P_{3} \times 25 \times 100$ and ${}^{4}P_{3} \times 25 \times 1000$

On adding all these sums, we get the sum of all the $4$ digit numbers formed by using the digits $1, 3, 5, 7, 9$ . Hence, the required sum is

${}^{4}P_{3} \times 25 \times 1 + {}^{4}P_{3} \times 25 \times 10 + {}^{4}P_{3} \times 25 \times 100 + {}^{4}P_{3} \times 25 \times 1000$

$= {}^{4}P_{3} \times 25 \times 1111$

$= 24 \times 25 \times 1111$

$= 6, 66, 600$ .

Find the sum of all $4$ digit numbers that can be formed using the digits $1, 3, 5, 7, 9$ .

Important Questions on Permutations and Combinations

Learn and Prepare for any exam you want !

Find the sum of all 4 digit numbers that can be formed using the digits 1, 3, 5, 7, 9.

Important Questions on Permutations and Combinations

Learn and Prepare for any exam you want !

Find the sum of all $4$ digit numbers that can be formed using the digits $1, 3, 5, 7, 9$ .