Permutations and Top Score

[rsweeney]

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My friend spent a lot of time helping me with the permutations and combinitorial questions in the Top Score math section, so I thought I would post how to do them on the forum. For me, I had no idea how to do them, but now I do so maybe I will at least help one person🙂

Permutations relate to questions of the type: What is the probability of rolling two fours with six rolls of one fair die. Here is the formula (we all love formulas--they make things such easier)

With the example above set up the problem this way:

1) (n, k)

n= total number of rolls=6
k= number of interest =2 (you want 2 fours)

SO, (6,2)

Pluge 6 and 2 into this equation (you'll love it ) 🙄


n!
------- X P(favorable)^k X P(unfavorable)^(n-k)
k!(k-n)!


(!) means factorial. Forexample 3! = 3X2X1
(P) means probability
(^) means exponent---"to the"
(X) means multiplied by---its not a variable
Therefore:

6!
---------- X (1/2)^2 X (5/6)^(4)
2!(6-2)!


And that is how you solve for the probability of getting 2(k) fours out of 6👎 rolls.

Now as for a combinitorial question like the one on Top Score which asks about how many cominations of three-groups of roses are possible if you have ten roses. Don't be fooled by different colored roses. To answer this question all you do is use the first expression in the equation above:

n!
------------
k!(n-k)!

This equation relates to the number of possible combinations.

So set it up:

n=10 for ten total roses
k=3 for groups of three

Therefore:

10!
--------- = 120 if I remember correctly.
3!(10-3)!


I know this s a random post but if it helps one person then its all worth it. I hope that when this window maximizes when people open it the equations and signs won't be messed up. I apoligize if they are 🙁

 

[lucyzhang]

wow, thanks rsweeney! it's so helpful

 

[lucyzhang]

btw, shouldn't the P(favored) in example1 be 1/6?

 

[rsweeney]

WOW, im sorry, the P(favored is 1/6) I must have glanced over the coin example. SORRY 😱 😱 😱 😱 😱

Thanks lucy

 

[rsweeney]

I figured that I should add that P(favored) + P(unfavored) should always equal 1.

P(favored) = P and P(unfavored) = Q

Therefore, P + Q =1

P(favored) with die = 1/6 (always!)
P(unfavored) with die is always 5/6


WITH COINS:

P(favored) always = 1/2
P(unfavored) always = 1/2

 

[critterbug]

This equation sure does help, but my problem is doin permutations in a timely manor. these type of problems get very time consuming especitally if the permutations are above 7. MAN, I miss my scientific calculator!!!

rsweeney,

Do you recommend just memorizing all the permuations up to 10!....Is that a good strategy? It seems that QR always leaves me rushing for time.

 

[rsweeney]

I have a feeling that even with permutations over 10, if you actually write out the factorials, they will cancel. Say the numerator is 10! and the denominator 20! The 10X9X8X7X6X5X4X3X2X1's cancel out and the equation is made simpler. I personally don't think the DAT will have any permutations on it at all. The one I took did not. You are correct in that they are too time consuming and it would be silly to memorize that equation. How could the ADA expect you to know it any way. Talk about high epectations if they do! I say that as long as you know the equations, you are one step closer to the right answer. remember, Top-Score is also guessing when the publishers write questions. The biology section, for example, on Top Score was way too detailed in terms of animal (invertebrate) physiology.