Question

A drawer contains n socks. When two are drawn randomly without replacement, the probability that both are red is 5/14. What is the smallest possible value of n

Answer 1

suppose there are x red socks among the n socks
so the prob of getting two reds
= (x/n)(x-1)/(n-1)
= x(x-1)/((n(n-1))

that is, numerator of our fraction consists of consecutive number multiples, and so does the denominator
... so it looks like 5/14 was reduced to lowest terms

let's build it up again, but looking at multiples of 5
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 ...
and those of 14
14 28 56 70 84 98 112 ....
5/14 = 10/28 or 15/42 or 20/56

ahh, 20/56 = (5x4)/(8x7)

So if there had been 8 socks of which 5 are red
prob(2red) = (5/8)(4/7) = 20/56 = 5/14

minimum value of n is 8