Video: Evaluating Permutations to Find the Value of an Unknown

Find the value of 𝑛 given that 𝑛𝑃4 = 24.

02:23

Video Transcript

Find the value of 𝑛 given that 𝑛P four equals 24.

We know that 𝑛Pπ‘Ÿ equals 𝑛 factorial over 𝑛 minus π‘Ÿ factorial and that 𝑛P four equals 24, which means we don’t know how many items our original set held, but we do know we’re selecting four of them. In cases where we aren’t given the 𝑛-value, it might not seem obvious where we should start, so let’s start by plugging in what we know into our formula, which will give us 𝑛P four equals 𝑛 factorial over 𝑛 minus four factorial. We know that 𝑛 factorial equals 𝑛 times 𝑛 minus one factorial. We can use this property to rewrite our numerator so that 𝑛 factorial is equal to 𝑛 times 𝑛 minus one factorial.

But that doesn’t get us any closer to simplifying the fraction. But we can expand 𝑛 minus one factorial, which would be 𝑛 minus one times 𝑛 minus one minus one factorial, which would be 𝑛 minus two factorial. And we could expand 𝑛 minus two factorial to be 𝑛 minus two times 𝑛 minus three factorial. And 𝑛 minus three factorial would be equal to 𝑛 minus three times 𝑛 minus four factorial. This allows us to cancel out the 𝑛 minus four factorial in the numerator and the denominator. And since we know that 𝑛P four equals 24, we can say that 24 will be equal to 𝑛 times 𝑛 minus one times 𝑛 minus two times 𝑛 minus three. This expression tells us we need four consecutive integers that multiply together to equal 24. And the 𝑛-value will be the starting point, the largest of those four values.

For smaller numbers like 24, we can try to solve this with a factor tree. 24 equals two times 12, 12 equals two times six, and six is two times three. So we can say 24 equals two times three times four. But remember, for this problem, we need four consecutive integers that multiply together to equal 24 and not three. One, two, three, and four are four consecutive integers, and when multiplied together, they equal 24. Remember that our 𝑛-value is the largest value. If we let 𝑛 equal four, 24 does equal four times three times two times one and confirms that 𝑛 equals four.

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