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# COUNTING METHODS

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SOLVE AND VERIFY

a.   (n-2)! / (n-3)! = 5

b.   (n+2)! / n! = 42

c.   (n-3)! / 2!(n-4)! = 17

d.   nP2 = 12

May 25, 2020

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I will do the first two problems and then I will let you apply the techniques I demonstrated for the rest of them.

a) $$\frac{(n-2)!}{(n-3)!}=5$$

The key to this question is to manipulate the left-hand side of the equation by "expanding" one of the factorials until a major cancellation occurs. Here is the way I would approach it.

$$\frac{(n-2)(n-3)!}{(n-3)!}=5$$

This manipulation is very helpful as it is now clear that $$(n-3)!$$ is a factor of both the numerator and the denominator. We can cancel this common factor to simplify the equation significantly.

$$n-2=5\\ n=7$$

Okay, let's apply this approach to the next one, too.

b) $$\frac{(n+2)!}{n!}=42$$

The procedure is the same: Expand one of the factorial terms until a major cancellation occurs.

$$\frac{(n+2)(n+1)n!}{n!}=42$$

This time, $$n!$$ was the common factor!

$$(n+2)(n+1)=42$$

This is a fairly standard quadratic. Let's solve it now.

$$n^2+n+2n+2=42\\ n^2+3n-40=0\\$$

This quadratic happens to be factorable.

$$(n-5)(n+8)=0\\ n=5\text{ or }n=-8$$

Always quickly check to make sure that your solutions are valid. Here, $$n=-8$$ would be considered an extraneous solution because the input of a factorial function should always be nonnegative.

May 25, 2020
edited by TheXSquaredFactor  May 25, 2020