Question Video: Using the Properties of Permutations to Simplify a Ratio between Permutations | Nagwa Question Video: Using the Properties of Permutations to Simplify a Ratio between Permutations | Nagwa

# Question Video: Using the Properties of Permutations to Simplify a Ratio between Permutations Mathematics • Third Year of Secondary School

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Evaluate ₁₂₃𝑃₁₀/₁₂₂𝑃₉.

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### 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.

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