Number of Permutations/Examples/Numbers Greater than 200 from 1, 2, 3, 4

Example of Use of Number of Permutations

Let $N$ be the number of ways you can make a number greater than $200$ using the digits $1$, $2$, $3$ and $4$ no more than once each.

Then:

$N = 42$

Proof

From Number of Permutations, you can make $24$ numbers using all $4$ digits once each, and all will be over $200$.

You cannot make a number greater than $200$ using $2$ digits or fewer.

From Number of Permutations, you can make $\dfrac {4!} {\paren {4 - 3}!} = \dfrac {24} 1 = 24$ numbers using only $3$ digits.

$6$ of those begin with $1$, and so are not greater than $200$.

Hence:

$N = 24 + 24 - 6$

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Exercises $\text I$: $5$