Find the value of 𝑛 given that 𝑛P
four equals 24.
We know that 𝑛P𝑟 equals 𝑛
factorial over 𝑛 minus 𝑟 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.