Video: Applications of the Counting Principle (Product Rule) and Permutations

A company labels their products with codes that start with three English letters followed by eight non-zero digits. Which of the following represents the number of codes that can be created with no repetition of any letter or digit? [A] 3𝑃3 + 8𝑃8 [B] 26𝑃3 + 9𝑃8 [C] 3𝑃3 Γ— 8𝑃8 [D] 26𝑃3 Γ— 9𝑃8

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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?

Let’s 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, we’re going to recall that the number of ways we can order π‘Ÿ items from a set of 𝑛 with no repetition and where order matters is 𝑛 P π‘Ÿ. And it’s 𝑛 factorial over 𝑛 minus π‘Ÿ 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 we’re interested in choosing eight nonzero digits, we can choose any digit between one and nine inclusive. So we’re choosing eight digits from a total of nine. Once again, order matters and we’re not using any repetition. So to choose eight digits from nine, it’s 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. It’s 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.

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