### Video Transcript

A company labels their products
with codes that start with three English letters followed by eight nonzero
digits. Which of the following represents
the number of codes that can be created with no repetition of any letter or
digit? Is it (A) three P three plus eight
P eight? Is it (B) 26 P three plus nine P
eight? Is it (C) three P three times eight
P eight? Or is it (D) 26 P three times nine
P eight?

Letβs begin by considering the two
parts of the code. The first part consists of three
English letters. Then we have eight nonzero
digits. And to work out the number of ways
of choosing or ordering the three English letters, weβre going to recall that the
number of ways we can order π items from a set of π with no repetition and where
order matters is π P π. And itβs π factorial over π minus
π factorial.

We want to choose three letters
from a total of 26 in the English alphabet. Order matters; in other words, ABC
is not equal to BAC. And so the number of ways to choose
these is 26 P three. Then, if weβre interested in
choosing eight nonzero digits, we can choose any digit between one and nine
inclusive. So weβre choosing eight digits from
a total of nine. Once again, order matters and weβre
not using any repetition. So to choose eight digits from
nine, itβs nine P eight.

If the number of ways of choosing
the three English letters is 26 P three and the number of ways of choosing the eight
digits is nine P eight, then the counting principle tells us that the total number
of possibilities, the total number of codes, is the product of these. Itβs 26 P three times nine P
eight. And if we compare that to the
options given in our question, we see that the answer is (D). The total number of codes that can
be created with no repetition of any letter or digit is 26 P three times nine P
eight.