Two integers from 1 to 16 are chosen at random and their corresponding

Question

Dibujo.JPG Two integers from 1 to 16 are chosen at random and their corresponding square in the above grid is shaded. What is the probability that these two shaded squares form a rectangle? a. 1/10 b. 1/8 c. 1/6 d. 1/5 e. 1/4

skovinsky · Answer

Hi!

Let's start with the formula that will govern the question (it's always a good idea to job down relevant formulae on your noteboard so you can see what you need to solve the problem):

Probability = # of desired outcomes/total # of possibilities.

The denominator is the easier part of the fraction to calculate. We have 16 total squares and we're choosing 2 of them, so we use the combinations formula:

16C2 = 16!/2!14! = 16*15/2 = 8*15 = 120

While there may be some fancy way to calculate the # of desired outcomes, i.e. how many pairs of squares actually form a rectangle, good ol' brute force is certainly going to be the quickest.

Looking at the diagram, we see that in each row, there are 3 pairs of rectangle-forming squares (e.g. in row 1, we can choose 1+2, 2+3 or 3+4). Since there are 4 rows, there are 4*3 = 12 horizontal rectangle-forming pairs.

Similarly, there are 3 pairs of rectangle-forming squares in each column (e.g. in column 1, we can choose 1+5, 5+9 or 9+13). Since there are 4 columns, there are 4*3 = 12 vertical rectangle-forming pairs.

So, the total number of rectangle-forming pairs is 12+12 = 24.

Our final calculation:

24/120 = 1/5... choose (D).

energetics · Answer

Calculating in a different way for practice:

Case 1 = 4 edge* 2 choices, all unique = 8
Case 2 = (8 side*3 choices)*1/2 = 12
1/2 of the choices are not unique (i.e on each side, there's 2 slots and 1st slot has 2 new choices and 2nd slot has 1 new choice) 3+3+3+3 = 12
Case 3 = (4 middle*4 choices)*1/4 = 4 
3/4 choices are not unique (i.e. 1st has 2 slots, 2nd has 1 slot, 3rd has 1 slot and 4th has 0 slots for new choices) 2+1+1+0 = 4

8+12+4 = 24/120 = 1/5

Bunuel  Is this correct? Also, how can we be sure in assuming we are choosing without replacement? The prompt doesn't say two  different  integers. Is it because it later says "two shaded squares"? Conceivably, couldn't we choose the same number and make a rectangle? This would add 16 to the total of viable choices, right?

eabhgoy · Answer

Total number of outcomes for selecting 2 numbers = 16 * 15 = 240
As we can select 16 numbers in the first go and 15 remaining numbers in the second.

Preferred Outcomes - 
We will need to see the placement of the numbers to see preferred outcomes:
4 corner numbers can be linked with 2 numbers each to form a rectangle. For e.g. 1 can be linked with 2 or 5. Hence 4*2 = 8
8 other numbers along the boundary ( 2, 3, 5, 8, 9, 12, 14, 15) can be linked with 3 numbers each. For e.g. 2 can be linked with 1, 3 or 6. Hence 8*3 = 24
4 numbers in the middle can be linked with 4 numbers each. For e.g. 6 can be linked with 2, 5, 7 or 10. Hence 4*4 = 16

Preferred Outcomes = 8 + 24 + 16 = 48

Probability = 48/240 = 1/5

Hence D is the correct answer.

tinytiger · Answer

Another way to look at the problem would be:

1) Number of ways to select 2 squares such that they are NOT in the same row or column = (16C1*9C1)/2! = 72 ways
2) Number of ways to select 2 squares such that they are IN the same row or column = 16C2 - 72 = 48 ways
3) We notice that only 1/2 of the time that when the squares are in the same row or column, they are next to each other i.e. they form a rectangle. Therefore, 48/2 = 24 ways.
4) Total probability = 24/120 = 1/5 (Answer: D)

bumpbot · Answer

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