Question Video: Evaluating the Ratio between Permutations Mathematics

Evaluate 123š‘ƒ10/122š‘ƒ9.

02:06

Video Transcript

Evaluate 123P10 over 122P nine.

We know to calculate š‘›Pš‘Ÿ, we have š‘› factorial over š‘› minus š‘Ÿ 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 š‘› factorial equals š‘› times š‘› 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 š‘›Pš‘Ÿ is equal to š‘› times š‘› minus one Pš‘Ÿ 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.

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