Problem D and Solution

Number Display

Helena’s Hardware Store is clearing out a particular style of single digits that are used for house numbers. There are currently only five \(5\)s, four \(4\)s, three \(3\)s, and two \(2\)s left.

How many different three-digit house numbers can be made using these single digits?

**Solution 1**

Let’s suppose that there were three or more \(2\)s available. For the first digit, the customer could choose from the digits \(5,~ 4,~ 3,\) and \(2\). Therefore, there would be \(4\) choices for the first digit. Similarly, there would be \(4\) choices for the second digit, and \(4\) choices for the third digit. This would give \(4\times 4 \times 4 = 64\) possible three-digit house numbers that could be made.

However, there are actually only two \(2\)s available, so not all of these house numbers can be made. In particular, the house number \(222\) cannot be made, but all others can.

Therefore, \(64 - 1 = 63\) different three-digit house numbers can be made using these single digits.

**Solution 2**

Let’s look at three different cases.

**Case 1:** All three digits in the house
number are the same

The house number could then be \(555\),
\(444\), or \(333\). The number \(222\) cannot be made since only two \(2\)s are available. Therefore, there are
\(3\) three-digit house numbers with
all three digits the same.

**Case 2:** Two digits are the same and the
third digit is different

There are \(4\) choices for the digits
that are the same, namely \(5,~ 4,~
3,\) and \(2\). For each of
these possible choices, there are \(3\)
choices for the third different digit. For example, if two of the digits
are \(5\), then the third digit could
be \(4,~ 3,\) or \(2\). Therefore, there are \(4\times 3 = 12\) ways to choose the digits.
For each of these choices, there are \(3\) ways to arrange the digits. For
example, suppose the digits are \(a,~a,\) and \(b\). The house number could be \(aab,~aba,\) or \(baa\).

Therefore, there are \(12\times 3 =
36\) three-digit house numbers with two digits the same and one
different.

**Case 3:** All three digits are
different

The customer has \(4\) choices for the
first digit, namely \(5,~ 4,~ 3,\) or
\(2\). Once that digit is chosen, there
are \(3\) choices for the second digit.
Once the first and second digits are chosen, there are \(2\) choices for the third digit. Therefore,
there are \(4\times 3\times 2 = 24\)
three-digit house numbers with all three digits different.

Therefore, \(3 + 36 + 24 = 63\) different three-digit house numbers can be made using these single digits.