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Question Video: Applications of the Counting Principle (Product Rule) Mathematics • 12th Grade

Liam must create a password for his new computer. The password is not case sensitive and consists of 4 English letters. Determine how many different passwords can be created if letters cannot be repeated.

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Video Transcript

Liam must create a password for his new computer. The password is not case sensitive and consists of four English letters. Determine how many different passwords can be created if letters cannot be repeated.

In order to answer this question, we need to use our knowledge of permutations. A permutation is an arrangement of a collection of items with no repetition and where order matters. The notation for this is ๐‘› P ๐‘Ÿ, where ๐‘Ÿ is the number of items being selected and ๐‘› is the total number of items. This can be calculated using the formula ๐‘› factorial divided by ๐‘› minus ๐‘Ÿ factorial.

Liam is selecting from the letters in the English alphabet. Therefore, ๐‘› is equal to 26. As he needs four of them for his password, ๐‘Ÿ is equal to four. This means that we need to calculate 26 P four. Liam needs to select four items from a group of 26 with no repetition and where order matters. This is equal to 26 factorial divided by 22 factorial as 26 minus four is equal to 22.

We know that ๐‘› factorial can be rewritten as ๐‘› multiplied by ๐‘› minus one factorial. This means that we can rewrite 26 factorial as 26 multiplied by 25 multiplied by 24 multiplied by 23 multiplied by 22 factorial. We can then divide the numerator and denominator by 22 factorial. This means that we are left with the product of the integers from 26 down to 23.

Multiplying these four values gives us 358,800. This is the total number of different passwords that Liam can create if the letters are not repeated. An alternative method here would be just to use a scientific calculator. Typing 26 followed by the ๐‘› P ๐‘Ÿ button followed by four and then pressing equals gives us an answer of 358,800.

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