Autor Tema: probability

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08 Noviembre, 2019, 07:36 am
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jacks

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Q: The sum of \( 3 \) positive integer is \( 20. \) Then the probability that they form a Triangle is

i am assuming \( a+b+c=20,a,b,c \in \mathbb{N} \) using wlog \( a\leq b \leq c \)

for triangle \( a+b\geq c\Rightarrow a+b+c \geq 2c\Rightarrow c\leq 10 \) and \( a+b+c\leq 3c\Rightarrow 20\geq 3c\Rightarrow c\geq 7 \)

so range of \( 7\leq c\leq 10 \)

For \( c=7. \) we have \( a+b=13, \) Then ordered pairs \( (6,7) \)

For \( c=8. \) we have \( a+b=12, \) Then ordered pairs \( (6,6),(5,7),(4,8) \)

For \( c=9. \) we have \( a+b=11, \) Then ordered pairs \( (5,6),(4,7),(3,8),(2,9) \)

For \( c=10. \) we have \( a+b=10, \) Then ordered pairs \( (5,5),(4,6),(3,7),(2,8),(1,9) \)

ordered pairs to form a triangle is \( 13 \)(favorable ways)

How do i calculate Total ways. please see it Thanks

08 Noviembre, 2019, 11:27 am
Respuesta #1

Luis Fuentes

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Hola

Q: The sum of \( 3 \) positive integer is \( 20. \) Then the probability that they form a Triangle is

i am assuming \( a+b+c=20,a,b,c \in \mathbb{N} \) using wlog \( a\leq b \leq c \)

for triangle \( a+b\geq c\Rightarrow a+b+c \geq 2c\Rightarrow c\leq 10 \) and \( a+b+c\leq 3c\Rightarrow 20\geq 3c\Rightarrow c\geq 7 \)

so range of \( 7\leq c\leq 10 \)

For \( c=7. \) we have \( a+b=13, \) Then ordered pairs \( (6,7) \)

For \( c=8. \) we have \( a+b=12, \) Then ordered pairs \( (6,6),(5,7),(4,8) \)

For \( c=9. \) we have \( a+b=11, \) Then ordered pairs \( (5,6),(4,7),(3,8),(2,9) \)

For \( c=10. \) we have \( a+b=10, \) Then ordered pairs \( (5,5),(4,6),(3,7),(2,8),(1,9) \)

ordered pairs to form a triangle is \( 13 \)(favorable ways)

How do i calculate Total ways. please see it Thanks

You can compute the number of all (unordered) triplets which form a triangle.

You have \( 5 \) ordered triples of type \( (a,a,b) \). There are \( 3 \) ways of rearrange it.

And you have \( 8 \) ordered triples of type \( (a,b,c) \). There are \( 6 \) ways of rearrange it.

So there are \( 5\cdot 3+8\cdot 6=63 \) triplets (a,b,c) providing a triangle.

The total ways are the number of integer solutions of:

\( x+y+z=20 \) with \( x,y,z\geq 1 \)

This is equivalent to the number of non negative integer solutions of:

\( a+b+c=17 \)


that is \( \displaystyle\binom{3+\color{red}17\color{black}-1}{3-1} \)

Best regards.

CORRECTED

08 Noviembre, 2019, 12:08 pm
Respuesta #2

Richard R Richard

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  • Oh Oh!!! me contestó... y ahora qué le digo...
abc\( \triangle \)vs\( \cancel{\triangle} \)order
1118\( \cancel{\triangle} \)3
2216\( \cancel{\triangle} \)3
3314\( \cancel{\triangle} \)3
4412\( \cancel{\triangle} \)3
5510\( \triangle \)3
668\( \triangle \)3
776\( \triangle \)3
884\( \triangle \)3
992\( \triangle \)3
1217\( \cancel{\triangle} \)6
1316\( \cancel{\triangle} \)6
1415\( \cancel{\triangle} \)6
1514\( \cancel{\triangle} \)6
1613\( \cancel{\triangle} \)6
1712\( \cancel{\triangle} \)6
1811\( \cancel{\triangle} \)6
1910\( \triangle \)6
2315\( \cancel{\triangle} \)6
2414\( \cancel{\triangle} \)6
2513\( \cancel{\triangle} \)6
2612\( \cancel{\triangle} \)6
2711\( \cancel{\triangle} \)6
2810\( \triangle \)6
3413\( \cancel{\triangle} \)6
3512\( \cancel{\triangle} \)6
3611\( \cancel{\triangle} \)6
3710\( \triangle \)6
389\( \triangle \)6
4511\( \cancel{\triangle} \)6
4610\( \triangle \)6
479\( \triangle \)6
569\( \triangle \)6
578\( \triangle \)6

There is not more ...I think so..
without ordering

\( P_{\triangle} =\dfrac{13}{33} \)


with order and rotation

\( P_{\triangle} =\dfrac{3\cdot 5+6\cdot 8}{3\cdot 9+6\cdot 24}=\dfrac{63}{171}=\dfrac{7}{19} \)
Saludos  \(\mathbb {R}^3\)

11 Noviembre, 2019, 03:22 pm
Respuesta #3

jacks

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Thanks Admin and Richard.

12 Noviembre, 2019, 07:55 am
Respuesta #4

Luis Fuentes

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Hi

Thanks Admin and Richard.

Note that I had a mistake. It has been corrected.

The total ways are the number of integer solutions of:

\( x+y+z=20 \) with \( x,y,z\geq 1 \)

This is equivalent to the number of non negative integer solutions of:

\( a+b+c=17 \)


that is \( \displaystyle\binom{3+\color{red}17\color{black}-1}{3-1} \)

Best regards.

CORRECTED

Best regards.