2 numbers out of 16 numbers can be chosen in 120 ways 16c2=120
The table it symmetrical:
1 2 3 4 are same with 13 14 15 16 ( Top Row & Bottom Row )
and
5 6 7 8 are same with 9 10 11 12 ( Middle Rows )
1 and 4 will be a rectangle with 2, 5 and 3,8 = 4 total
2 and 3 will be a rectangle with 1,6,3 and 2,7,4 = 6 total
also you should keep in mind that these rectangles overlap!
so 6+4=10
5 and 8 will be a rectangle with 1,6,9 and 4,7,12 = 6 total
6 and 7 will be a rectangle with 2,5,7,10 and 3,6,11,8 = 8 total
so 8+6=14
Since the rest of the matrix is simetrical and we overlap all the rectangles:
10+14=24 total
24/120 = 1/5
It takes time to explain and write it but this can be solved under 2 minutes.
Probability Problem
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cramya
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Similar approach
Picking 2 numbers out 16 16C2 = 120
LISTING OUT THE POSSIBILITES BY BRUTE FORCE APPROACH
1:2,1:5,2:3,2:6,3:4,3:7,4:8,5:6,5:9,6:7,6:10,7:8,7:11,8:12,9:10,9:13,10:11,10:14,11:12,11:15,12:16,13:14,14:15,15:16
=24
So 24/120 = 1/5
Friendly suggestion :If we are not able to find the shortcut method we are better off going wiht the brute force approach instead of wasting time trying to come up with a shorcut.
Picking 2 numbers out 16 16C2 = 120
LISTING OUT THE POSSIBILITES BY BRUTE FORCE APPROACH
1:2,1:5,2:3,2:6,3:4,3:7,4:8,5:6,5:9,6:7,6:10,7:8,7:11,8:12,9:10,9:13,10:11,10:14,11:12,11:15,12:16,13:14,14:15,15:16
=24
So 24/120 = 1/5
Friendly suggestion :If we are not able to find the shortcut method we are better off going wiht the brute force approach instead of wasting time trying to come up with a shorcut.
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logitech
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Cramya,cramya wrote:Similar approach
Picking 2 numbers out 16 16C2 = 120
LISTING OUT THE POSSIBILITES BY BRUTE FORCE APPROACH
1:2,1:5,2:3,2:6,3:4,3:7,4:8,5:6,5:9,6:7,6:10,7:8,7:11,8:12,9:10,9:13,10:11,10:14,11:12,11:15,12:16,13:14,14:15,15:16
=24
So 24/120 = 1/5
Friendly suggestion :If we are not able to find the shortcut method we are better off going wiht the brute force approach instead of wasting time trying to come up with a shorcut.
Tell us more about Brute Force Approach master!
I am all ears!
LGTCH
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"DON'T LET ANYONE STEAL YOUR DREAM!"
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"DON'T LET ANYONE STEAL YOUR DREAM!"
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cramya
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Serioulsy the master stuff creeps me out 
Jus kidding!
I was just saying that listing out the possibiliteis works sometimes (may not work if there are 1000's of possibilities) In this problem it just happenned to work wihtin the 2 minutes we have to solve a problem!
Strictly my opinion. I think ur approach is good but not all minds are alike!! Whatever works!
Jus kidding!I was just saying that listing out the possibiliteis works sometimes (may not work if there are 1000's of possibilities) In this problem it just happenned to work wihtin the 2 minutes we have to solve a problem!
Strictly my opinion. I think ur approach is good but not all minds are alike!! Whatever works!
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uttara
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I did it this way using the attached image:
Two numbers can form a rectangle, only if one of their sides is common. There are three methods of choosing such numbers. Each RED number if chosen first can have 2 numbers which will help it make a rectangle. Similarly each BLUE number can have 3 such numbers and each BLACK number can have 4 such numbers.
From the table
Total probability = probability when the first selection is a Red number + probability when the first selection is a Blue number + probability when the first selection is a Black number
= (4/16*2/15)+(8/16*3/15)+(4/16*4/15)
=1/30+1/10+1/15
= (1+3+2)/30=6/30=1/5
For ease of understanding:
probability when the first selection is a Red number
if first number is a RED one, then there are only TWO numbers which if chosen next can make a rectangle eg. 1 with 2, 5 & 4 with 3,8 & 13 with 9,14 & 16 with 12, 15
ie probability of choosing method 1 = Probability of choosing a RED no from sixteen numbers * Probability of choosing the two numbers from fifteen which will form the rectangle
= (4/16)*(2/15)
probability when the first selection is a Blue number
if first number is a BLUE one, then there are only THREE numbers which if chosen next can make a rectangle eg. 2 with 1,3,6 & 3 with 2,7,4 etc.
= (8/16*3/15)
probability when the first selection is a Black number
if first number is a BLACK one, then there are only FOUR numbers which if chosen next can make a rectangle eg. 6 with 2,5,10,7 etc
= (4/16*4/15)
to get the final probability, just add up these three probabilities.
Two numbers can form a rectangle, only if one of their sides is common. There are three methods of choosing such numbers. Each RED number if chosen first can have 2 numbers which will help it make a rectangle. Similarly each BLUE number can have 3 such numbers and each BLACK number can have 4 such numbers.
From the table
Total probability = probability when the first selection is a Red number + probability when the first selection is a Blue number + probability when the first selection is a Black number
= (4/16*2/15)+(8/16*3/15)+(4/16*4/15)
=1/30+1/10+1/15
= (1+3+2)/30=6/30=1/5
For ease of understanding:
probability when the first selection is a Red number
if first number is a RED one, then there are only TWO numbers which if chosen next can make a rectangle eg. 1 with 2, 5 & 4 with 3,8 & 13 with 9,14 & 16 with 12, 15
ie probability of choosing method 1 = Probability of choosing a RED no from sixteen numbers * Probability of choosing the two numbers from fifteen which will form the rectangle
= (4/16)*(2/15)
probability when the first selection is a Blue number
if first number is a BLUE one, then there are only THREE numbers which if chosen next can make a rectangle eg. 2 with 1,3,6 & 3 with 2,7,4 etc.
= (8/16*3/15)
probability when the first selection is a Black number
if first number is a BLACK one, then there are only FOUR numbers which if chosen next can make a rectangle eg. 6 with 2,5,10,7 etc
= (4/16*4/15)
to get the final probability, just add up these three probabilities.
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brianoduor
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No of rectangles: 16C2
No of rectangles per row: 3
No of rows: 4
Flip the square to double the combinations:
total: 3*4*2 = 24
Probability: 24/120
No of rectangles per row: 3
No of rows: 4
Flip the square to double the combinations:
total: 3*4*2 = 24
Probability: 24/120
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logitech
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NASTY!!!!brianoduor wrote:No of rectangles: 16C2
No of rectangles per row: 3
No of rows: 4
Flip the square to double the combinations:
total: 3*4*2 = 24
Probability: 24/120
Now we are talking! you are the man! This is how it is done ladies and gentleman!
LGTCH
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"DON'T LET ANYONE STEAL YOUR DREAM!"
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"DON'T LET ANYONE STEAL YOUR DREAM!"
