Number of Permutations/Examples/Even Integers from 1, 2, 3, 4

Example of Number of Permutations

Let $N$ be the number of even integers which can be made using all the digits $1$, $2$, $3$ and $4$.

Then:

$N = 12$

Proof

An integer formed using the digits $1$, $2$, $3$ and $4$ is even if and only if it ends in $2$ or $4$.

Those $4$ digit integers ending in $2$ consist of the $3$ digits integers that can be made with $1$, $3$ and $4$

Those $4$ digit integers ending in $4$ consist of the $3$ digits integers that can be made with $1$, $2$ and $3$

From Number of Permutations, the total number of integers which can be made using $3$ different digits is $3!$.

$N = 2 \times 3! = 12$

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Exercises $\text I$: $6$