Find the numbers between $1500$ and $5000$ that can be made from the digits $1,2,4,5,7$ and $8$ if each digit is only used once.

Question

Find the numbers between $1500$ and $5000$ that can be made from the digits $1,2,4,5,7$ and $8$ if each digit is only used once.

Can someone show the steps?

Answer 1

You have to add up a few cases:

1. 15AB
2. 1ABC (where A = 7 or A = 8)
3. ABCD (where A = 2 or A = 4)

And per case count how many ways you can select (valid) combination.

It's just manual labor.

(Given that 1 and 2 can be combined:

1. $1ABC$, where $A \in \{5,7,8\}$ and select 2 from $\{2,4,a,b\}$ gives $1*3*4*3 = 36$ numbers. $a,b$ are the leftovers after selecting $A$ from $5,7,8$.

2. $ABCD$, where $A \in \{2, 4\}$ and select 3 from $\{1,5,7,8,c\}$ gives $2*5*4*3 = 120$ numbers

In total: $36 + 120 = 156$.