Question Video: Permutations and Factorials | Nagwa Question Video: Permutations and Factorials | Nagwa

Question Video: Permutations and Factorials Mathematics • Third Year of Secondary School

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If (2𝑛 + 1) 𝑃₆ : (2𝑛 βˆ’ 1) 𝑃₅ = 272 : 11, find 𝑛!.

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Video Transcript

If the ratio of two 𝑛 plus one 𝑃 six to two 𝑛 minus one 𝑃 five is equal to 272 to 11, find 𝑛 factorial.

We recall that when dealing with permutations, the notation π‘›π‘ƒπ‘Ÿ is equal to 𝑛 factorial divided by 𝑛 minus π‘Ÿ factorial. This means that two 𝑛 plus one 𝑃 six is equal to two 𝑛 plus one factorial divided by two 𝑛 plus one minus six factorial. The denominator simplifies to two 𝑛 minus five factorial. We recall that the factorial of any integer is the product of that integer and all the positive integers below it. 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two, and so on, multiplied by two multiplied by one.

This means that 𝑛 factorial is also equal to 𝑛 multiplied by 𝑛 minus one factorial. This means that we can rewrite two 𝑛 plus one factorial as two 𝑛 plus one multiplied by two 𝑛 multiplied by two 𝑛 minus one multiplied by two 𝑛 minus two multiplied by two 𝑛 minus three multiplied by two 𝑛 minus four multiplied by two 𝑛 minus five factorial. We can then cancel this last term with the denominator. This leaves us with an expression for two 𝑛 plus one 𝑃 six. We can repeat this for two 𝑛 minus one 𝑃 five. This is equal to two 𝑛 minus one factorial divided by two 𝑛 minus six factorial. This time, we can cancel two 𝑛 minus six factorial.

We’re dealing with the ratio of these two terms. And we notice they have four common factors: two 𝑛 minus one, two 𝑛 minus two, two 𝑛 minus three, and two 𝑛 minus four. We can simplify any ratio by dividing by common factors. We can now rewrite this so that the ratio of two 𝑛 plus one 𝑃 six to two 𝑛 minus one 𝑃 five is equal to two 𝑛 plus one multiplied by two 𝑛 to two 𝑛 minus five. We are also told in the question that this ratio is equal to 272 to 11.

We can now rewrite these ratios as fractions. Two 𝑛 multiplied by two 𝑛 plus one divided by two 𝑛 minus five is equal to 272 over 11. We can cross multiply so that 22 𝑛 multiplied by two 𝑛 plus one is equal to 272 multiplied by two 𝑛 minus five. Expanding our brackets or distributing the parentheses gives us 44𝑛 squared plus 22𝑛 is equal to 544𝑛 minus 1360. Subtracting 544𝑛 and adding 1360 to both sides gives us the quadratic equation 44𝑛 squared minus 522𝑛 plus 1360 is equal to zero. We can then divide both sides of this equation by two, giving us 22𝑛 squared minus 261𝑛 plus 680 equals zero.

Whilst it is not immediately obvious, this quadratic expression can be factored. 22𝑛 squared minus 261𝑛 plus 680 is equal to 22𝑛 minus 85 multiplied by 𝑛 minus eight. We could check this by redistributing the parentheses and simplifying our expression. As the product of these two parentheses is zero, one of the parentheses themselves must be equal to zero. Either 22𝑛 minus 85 equals zero or 𝑛 minus eight equals zero. Solving these two equations gives us 𝑛 is equal to 85 over 22 and 𝑛 is equal to eight. 𝑛 must be a positive integer, which rules out our first answer. As 𝑛 is therefore equal to eight, we need to calculate eight factorial. This is equal to eight multiplied by seven multiplied by six, and so on, all the way down to one. This is equal to 40320.

If the ratio of two 𝑛 plus one 𝑃 six to two 𝑛 minus one 𝑃 five is equal to 272 to 11, then 𝑛 factorial is 40320. We could check this answer by calculating 17𝑃 six and 15𝑃 five and check that they are in the ratio 272 to 11.

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