Video Transcript
In a gallery, each painting is
referenced by two distinct English letters and a four-digit number which has no
zeros and no repeated digits. How many paintings can be
referenced using this system?
In order to answer this question,
we begin by splitting it into two parts. Firstly, the reference number
contains two distinct English letters. So we’re going to be considering
the 26 letters in the English alphabet. And secondly, we need to consider
all the four-digit numbers that have no zeros and no repeated digits. We begin by recalling the product
rule for counting. This states that to find the total
number of outcomes for two or more events, we multiply the number of outcomes of
each event.
Let’s consider the total number of
ways of ordering our English letters. Since there are 26 letters in the
alphabet, there are 26 ways of finding the first letter. As the two letters have to be
distinct, once we’ve chosen that first letter, we only have 25 left to choose
from. This means that the total number of
ways of choosing our English letters is 26 multiplied by 25, which is equal to
650.
Next, we need to consider the
four-digit numbers. As we can use any of the numbers
from one to nine, there are nine possible ways of choosing our first digit in our
four-digit number. Since the digits cannot be
repeated, there are then eight possible ways of selecting the second digit. In the same way, we then have seven
to choose from for the third digit. And finally, for the fourth digit,
we have six to choose from. The total number of ways of
choosing our four-digit number is therefore equal to nine multiplied by eight
multiplied by seven multiplied by six, which is equal to 3,024.
Our final step is to combine these
sets of outcomes, the number of ways of ordering our letters and the number of ways
of ordering our number. We need to multiply 650 by
3,024. Typing this into our calculator, we
have 1,965,600. And we can therefore conclude that
this is the number of paintings that can be referenced using this system. The gallery can reference 1,965,600
paintings.
