A sock drawer contains 36 pairs of socks, which are colored white, bro

Question

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A sock drawer contains 36 pairs of socks, which are colored white, brown, or black, in some combination as follows:

• 5 pairs are colored with all three colors;
• 25 pairs have some white;
• 28 pairs have some brown;
• 20 pairs have some black.

In the table, select the number of pairs that have only one color and the number of pairs that have exactly two colors. Make only two selections, one in each column.

chetan2u · Accepted Answer

A very very quick logical answer to the question.

Since there are five pairs with all three colours, remaining 36-5 or 31 have exactly one or exactly two.

Look at the options that add up to 31. It is 4+27. 
80 to 90% of the battle is already won.

A simple glance at numbers 25, 28 and 20 result of just 36 pairs of socks will tell that there will be many more with two colours than single colour.

Answer is 
Single: 4
Exactly two colours: 27

Posted from my mobile device

brp616 · Answer

We know there are 36 pairs of socks

total 'colors' = 25 + 28 + 20 = 73

If 5 socks have all 3 colors, we know 5 pairs are counted in all 3 buckets, and we should de-duplicate by subtracting 5 * 2 = 10 - so we get 63 pairs...

But wait! this is much more than our total of 36 socks. So we need to find how many are still double counted. If a sock has 2 colors it is counted twice in our 63 color total, so to get the number of 2-color pairs we can do total colors - total socks = 63 - 36 = 27 - we know 27 pairs have 2 colors, and that gives us our answer for that part.

For the number of 1 color socks, we can do total - 3 color - 2 color or 36 - 27 - 5 = 36 - 32 = 4 pairs!

And this makes sense - if we have 73 total colors, we can do 4 + 27*2 (27 pairs w 2 colors) +5*3 (5 pairs w 3 colors) = 4 + 54 + 15 = 73

Purnank · Answer

Problem type overlapping set,
lets assume number of E1 = exactly 1 colored pair, E2 = exactly 2 colored pair, E3 = exactly 3 colored pair,
Given - A sock drawer contains 36 pairs of socks, which are colored white, brown, or black, in some combination as follows:

• 5 pairs are colored with all three colors; = E3 = 5
• 25 pairs have some white;
• 28 pairs have some brown;
• 20 pairs have some black.

Total pair of socks = 36 = E1 + E2 + E3,
E1 + E2 = 36 - 5 = 31 --- Eq1
and, E1 + 2E2 + 3E3 = 25 + 28 + 20 = 73
E1 + 2E2 = 73 - 15 = 58 ---- Eq2
on solving E2 = 27
E1 = 4

siddhantvarma · Answer

This is a good solution! I understand everything in here except the fact that how does subtracting the number of 2-color pairs from total socks ( 63 - 36 = 27 ) gives us the number of 2 color socks?
@ KarishmaB   @  Bunuel  Can you please explain this? I'm also curious how one would solve this using Venn diagrams, if possible at all.

Akshaynandurkar · Answer

from figure, we can write that
73-2(x+y+z)+(x+y+z)+5 = 36
(x+y+z) = 27 (2 colored socks)

for one colored socks
(20-x-y)+(23-x-z)+(15-y-z) = 58-2*27 = 4

siddhantvarma · Answer

Thanks! This was helpful. I tried solving this question using Venn diagram and got stuck because I took 25 as pairs with  ONLY  white,
28 as pairs with  ONLY  some brown and 20 pairs with  ONLY  black.

egmat · Answer

When we add up all the individual color counts:
 25  (white) +  28  (brown) +  20  (black) =  73

But we only have  36  pairs! Where did the extra  37  come from?

Here's what happens when you count this way: 
- A  one-color  pair (say, all white) gets counted  1  time - correct!
- A  two-color  pair (say, white-brown) gets counted  2  times - once in white, once in brown. That's  1  extra count.
- A  three-color  pair gets counted  3  times - once in each color. That's  2  extra counts.

So the "overcount" of 37 comes from:

Overcount = (two-color pairs) x  1  + (three-color pairs) x  2

We know three-color =  5 , so:

37  = (two-color) +  5  x  2 
 37  = (two-color) +  10 
 Two-color = 27

And since total = one-color + two-color + three-color:
 36  = one-color +  27  +  5 
 One-color = 4

Answer: One color = 4, Two colors = 27

Regarding Venn Diagrams: 
Yes, you can use a Venn diagram with  3  overlapping circles (White, Brown, Black). The center region where all three overlap =  5 . You'd then need to find the other  7  regions. However, the overcounting method above is faster since we don't need to find each individual region - we just need the totals for "exactly one color" and "exactly two colors."

Think of it this way:  The overcount tells us how many times we "double-dipped" when adding. Each two-color pair was dipped twice, each three-color pair was dipped three times. The extra dips =  37 , and working backwards gives us the answer.

Rickooreoisb · Answer

Important theory - https://gmatclub.com/forum/advanced-ove ... 44260.html

A sock drawer contains 36 pairs of socks, which are colored white, bro

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A sock drawer contains 36 pairs of socks, which are colored white, bro

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