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Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

FACTORIAL QUESTION?

Anyone know explain why:

n! / (n-2)! = 42

is equal to

n^2 = n + 42

7 Answers

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  • 1 month ago
    Favorite Answer

    n! / (n-2)! = 42

    n(n -1)(n - 2)! / (n - 2)! = 42

    n(n - 1) = 42

    n² - n = 42

    n² = n + 42

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  • Philip
    Lv 6
    1 month ago

    n! = n(n-1)(n-2)! Therefore n!/(n-2)! = n(n-1)(n-2)!/(n-2)! = n(n-1). Then n!/(n-2)! = 42

    becomes n(n-1) = 42, ie., n^2 -n -42 = 0, ie., (n+6)(n-7) = 0. Negative root n = -6 is

    extraneous since factorials, by definition, exist only for non-negative integers. Then 

    n = 7.

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  • 1 month ago

    n! / (n-2)! = 42

    Alternate form assuming n is real:

    n^2 = n + 42

    Alternate form:

    (n - 1) n = 42

    Solution:

    n = 7

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  • 1 month ago

    n! = n x (n-1) x (n - 2) x (n - 3) x (n - 4) x...

    (n - 2)! = (n - 2) x ( n-3) x ( n - 4) x ....

    By cancelling down the LHS you are left with

    n x ( n- 1) = 42

    n^2 - n = 42

    n^2 = n + 42

    Hope that helps!!!!

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  • Ian H
    Lv 7
    1 month ago

    n!/(n - 2)! = n(n – 1) * (n - 2)!/(n - 2)! = n^2 – n = 42 

    then n^2 = n + 42 or equivalently 

    n^2 - n – 42 = (n – 7)(n + 6) = 0 

    Positive option is n = 7 

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  • 1 month ago

    Before starting: example

    n! = a * b * c * d * e * f * g ← you can see that: n = g

    (n - 2)! = a * b * c * d * e

    So you can deduce that:

    n! = a * b * c * d * e * f * g

    n! = (a * b * c * d * e) * f * g

    n! = (n - 2)! * f * g → recall: g = n

    n! = (n - 2)! * f * n → so, you can say that: f = n - 1

    n! = (n - 2)! * (n - 1) * n

    Now, the equation:

    n! / (n - 2)! = 42 → recall the previous result

    (n - 2)! * (n - 1) * n / (n - 2)! = 42 → you can sylpmly by (n - 2)!

    n * (n - 1) = 42

    n² - n = 42

    n² - n + (1/2)² = 42 + (1/2)²

    n² - n + (1/2)² = (168/4) + (1/4)

    [n - (1/2)]² = 169/4

    n - (1/2) = ± 13/2

    n = (1/2) ± (13/2)

    n = (1 ± 13)/2 → only the positive value

    n = (1 + 13)/2

    n = 7

    Check it:

    7! = 1 * 2 * 3 * 4 * 5 * 6 * 7 = 5040

    (7 - 5)! = 5! = 120

    = 5040/120

    = 42

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  • 1 month ago

    That depends on what I is....nI=42(n-2)I

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