MitDavidDv wrote:
If two integers are chosen at random out of the set {2, 5, 7, 8}, what is the probability that their product will be of the form a^2 – b^2, where a and b are both positive integers?
A: 2/3
B: 1/2
C: 1/3
D: 1/4
E: 1/6
Shalom! I am currently studying the probability chapter of the
Manhattan GMAT Word Translations book. I am looking forward to the different outcomes and answers.
I just used the exhaustive method. Count everything that fits.
2,5=10
2,7=14
2,8=16
5,7=35
5,8=40
7,8=56
Write down all perfect squares until 100
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Pick one number at a time.
10 -- Keep adding with every perfect square and see whether the result is also there in the set.
10+1=11(Not there)
10+4=14(Not there)
10+9=19(Not there)
10+16=26(Not there)
10+25=35(Not there)
we can stop here as the difference between all consecutive perfect squares after 35 will be more than 10.
10- Not possible to represented as a^2-b^2
Repeat the same with all the products;
14-Not Possible
16: Let's check this;
16+1=17(Not there)
16+4=20(Not there)
16+9=25(There in the set)
16- can be represented as a^2-b^2 i.e. 5^2-3^2
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Likewise:
35- 6^2-1^2
40- 7^2-3^2
56- 9^2-5^2
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In the sample set:
{10,14,16,35,40,56}
{10,14}- Not Possible: Count=2
{16,35,40,56}- Possible: Count=4
Total Count=6
P=Favorable/Total=4/6=2/3
Ans: "A"
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