Evaluate 123P10 over 122P nine.
We know to calculate 𝑛P𝑟, we have
𝑛 factorial over 𝑛 minus 𝑟 factorial, which means the numerator is 123 factorial
over 123 minus 10 factorial. And our denominator is 122
factorial over 122 minus nine factorial, which we simplify to be 123 factorial over
113 factorial all over 122 factorial over 113 factorial. If we rewrite this with division,
it looks like this. And we know dividing by a fraction
is multiplying by the reciprocal, which will be 123 factorial over 113 factorial
times 113 factorial over 122 factorial. And then we have 113 factorial in
the numerator and the denominator. So we have 123 factorial over 122
factorial. And we know that 𝑛 factorial
equals 𝑛 times 𝑛 minus one factorial, which means we could rewrite the numerator
as 123 times 122 factorial. The 122 factorial in the numerator
and the denominator cancel out, and this ratio simplifies to 123.
However, we could have saved
ourselves some work by remembering an additional property of permutations. And that is that 𝑛P𝑟 is equal to
𝑛 times 𝑛 minus one P𝑟 minus one. If we notice that 122 equals 123
minus one and nine equals 10 minus one, we could rewrite our numerator as 123 times
122P nine, which would mean we would have a permutation of 122P nine in the
numerator and in the denominator. And so the fraction would reduce to
123. When it comes to evaluating and
simplifying permutations, we always want to look for patterns that can lead us back
to properties which will help us simplify.