Video Transcript
Evaluate 123𝑃10 over 122𝑃
nine.
We’ve been given a fraction in
which the numerator and the denominator are both permutations in the form
𝑛𝑃𝑟. One way to solve this problem would
be to expand the numerator and the denominator. The numerator 123𝑃10 becomes 123
factorial over 123 minus 10 factorial. And then, we would divide that by
122 factorial divided by 122 minus nine factorial. Solving the subtraction, we can
simplify both of these expressions to 113 factorial. We know that dividing by a fraction
is the same thing as multiplying by the reciprocal. And our 113 factorial in the
numerator and the denominator cancel each other out. To do a bit more simplifying, we
can say that 𝑛 factorial is equal to 𝑛 minus one factorial. And that means we can rewrite 123
factorial as 123 times 122 factorial. In this form, we’re able to quickly
see that the 122 factorial in the numerator and the denominator cancel out, leaving
us with 123.
However, there was a more
straightforward way to solve this problem using a property of permutations. This tells us for any permutation
𝑛𝑃𝑟, it will be equal to 𝑛 times 𝑛 minus one 𝑃 𝑟 minus one. Therefore, 123𝑃10 is equal to 123
times 122𝑃 nine. Using this method of
simplification, we’d be able to remove the 122𝑃 nine from the numerator and the
denominator without expanding both of them to factorial form. In either case, we see that 123𝑃10
over 123𝑃 nine is equal to 123.
