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June 26th, 2013, 08:18 AM   #1
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Probability Involving Possible Outcomes of making a number.

A question was given such that


find the possible values of x , y , z ....given the variable MUST BE POSITIVE INTEGER. (ans : 21)

FINE , i did it by listing method.
from 1 + 1 + 6 , 1 + 6 + 1 , ...all the way to 3 + 2 + 2

but from the equation given , we can incur that this type of question is possible given the value must be min equal the amount of variable

such that x + y + z = 3 ......it cannot be 2 because decimal takes forever to calculate for a high schooler in an exam.

x + y + z = n , such that n larger or equal to the sum of total of variable.

is there a way to calculate the available number of outcomes??
i don't think it is easy to do listing method because exercise question is giving me longer equation like w + x + y + z = 17


--> this is what i have done so far.

assume x + y + z = 3 --> possible outcome = 1 because 3 divide by (3 variable) is 1.

x + y + z = 4 --> possible outcome is 3 ........because 4/3 = 1.xx round off to 2 we get 2 + 1 + 1 --> 3C1 = 3

x + y + z = 5 .... 5/3 = 1.67 round to 2.
1 + 2 + 2 ( 3 outcome )
1 + 1 + 3 ( 3 outcome )
total : 6 ( 3p3 = 6 ??)

x + y + z = 6 ..........6/3 = 2

2 + 2 + 2 ( 1 outcome )
4 + 1 + 1 ( 3 outcome )
3 + 2 + 1 ( 6 outcome )

total : 10 outcome. ( 1 + 2 + 3 + 4 = 10 ??)

i can't see the relationship of the progression.

is there a way to calculate the available number of outcomes??
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June 26th, 2013, 07:59 PM   #2
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Re: Probability Involving Possible Outcomes of making a numb

There is, in fact, a systematic method of doing this, but only under the assumption that the integers are unordered (i.e. 2+3+6 is the same as 6+2+3).
Let a+b+c=n. We can visually represent it below:

* * * * * * ...

where there are n "stars". We need to divide the stars into three groups (because there are three integers) with two lines, and there are n-1 gaps where are lines could be, so the number of ways to do this is just .

We could also do it with four integers: a+b+c+d=n which becomes

In general, if there are m integers a1, a2, ... a_m that sum to n, then there are ways of choosing the values of the m integers.

Note that this method only works with UNORDERED SUMS.
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June 27th, 2013, 12:03 AM   #3
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Re: Probability Involving Possible Outcomes of making a numb

can you use your above method to solve

x + y + z = 8 for me to see?

the algebra is not helping me to understand it......and i also forgot what is this notation means.
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June 29th, 2013, 07:31 PM   #4
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Re: Probability Involving Possible Outcomes of making a numb

Hello, rnck!

Quote:

Find the number of possible values of
given that the variables must be positive integers.
(Answer: 21)

Place eight objects in a row, leaving a space between them.
[color=beige]. . . . [/color]

Select 2 of the 7 spaces and insert "dividers".


Then:

[color=beige]. . . [/color]

[color=beige]. . . [/color]

[color=beige]. . . [/color]




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June 30th, 2013, 11:17 AM   #5
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Re: Probability Involving Possible Outcomes of making a numb

Quote:
Originally Posted by soroban
Hello, rnck!

Quote:

Find the number of possible values of
given that the variables must be positive integers.
(Answer: 21)

Place eight objects in a row, leaving a space between them.
[color=beige]. . . . [/color]

Select 2 of the 7 spaces and insert "dividers".


Then:

[color=beige]. . . [/color]4,\,1,\,3)" />

[color=beige]. . . [/color]1,\,5,\,2)" />

[color=beige]. . . [/color]1,\,3,\,4)" />




I recall you showing this once before and, as I did before, I just have to marvel at the brilliance of this visual demo.
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July 1st, 2013, 05:11 AM   #6
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Re: Probability Involving Possible Outcomes of making a numb

wow thats insane....how u decide where to put divider?

is n(divider) = n(variable)-1 ?

and after u put it , how come suddenly pop choosing 2 objects from 7 objects?

7! seems understandable but the denominator 2!5! come from where?

you gotta tell me man! my whole school even my teach cant formulate out this piece of math!!
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July 1st, 2013, 07:21 AM   #7
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Re: Probability Involving Possible Outcomes of making a numb

w + x + y + z = 17

if there is 16 gap , and 3 divider...



indeed it is answer.....

-Edit : ......i had understand how it works. can close topic le.

Indeed , your illustration is MARVELOUS.....it looks so familiar i suddenly realised i've been tricked.
I never would have thought that a simple question of asking how to arrange 2 red chair in between 8 green chair such that red chair must be placed in between 2 green chair can be asked in this way........
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