Hi Could you re-explain the green sock bit? So out of 4 socks, we first check for 2 pairs (which is 1/90). Then what is the scenario for green socks? Thank you!

Hello nancyjose,
try this approach if it makes it easier to understand-

We have 3 pairs in total i.e 2red, 2 blue and 2 green

and 6 number of socks in total.

Now Select 2 pairs out of 3 i.e = 3c2 = 3 (since we need exactly 2pair)

Selecting 4 socks randomly = 6c4 = 15

Probability = 3/15 = 1/5

We have three scenarios of two matching pairs: 1) a red pair and a blue pair; 2) a red pair and a green pair; 3) a blue pair and a green pair. Let’s start with the probability of selecting a red pair and a blue pair. To select a red pair and a blue pair is to select two red socks and two blue socks. So let’s assume the first two socks are red and the last two socks are blue; the probability of selecting these socks in that order is:

P(R, R, B, B) = 2/6 x 1/5 x 2/4 x 1/3 = 1/6 x 1/5 x 1/3 = 1/90.

However, the two red socks and the two blue socks, in any order, can be selected in 4!/(2! x 2!) = 24/4 = 6 ways. Thus, the probability of two red socks and two blue socks is:

P(2R and 2B) = 1/90 x 6 = 6/90 = 1/15.

Using similar logic, we see that the probability of pulling a red pair and a green pair is 1/15, and so is the probability of pulling a blue pair and a green pair. Thus, the total probability is:

1/15 + 1/15 + 1/15 = 3/15 = 1/5.

Alternate Solution:

From a total of 6 socks, two pairs, i.e., 4 socks, can be pulled in 6C4 = 6!/(4! 2!) = (6 x 5)/2 = 3 x 5 = 15 ways.

Three of these choices contain two matching pairs, namely: 1) a red pair and a blue pair, 2) a blue pair and a green pair; 3) a red pair and a green pair.

Therefore, the probability of pulling two matching pairs is 3/15 = 1/5.

Answer: A

Theres a few ways to approach this, all very simply if you know what to look for.

method #1,
the probtability of selecting any sock is 1
the probability of selecting its partner is 1/5
the probability of selecting any other sock is 1
the probability of selecting its partner is 1/3.
1*1/5*1*1/3=1/15
but remember, there are 3 different color socks, so you just multiply
3*1/15= 3/15= 1/5

Method number 2
order doesn't matter here, it usually only matters with coin tosses and elections, so we can use nCr.
the total number of combinations of socks is 6c4= 6!/(2!*4!)=15, this is our denominator
There are 3c2 =3 ways of selecting matching socks,
(# of matching pair combos)/total combos= 3/15 = 1/5

the answer is A

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