1. ## Counting and combination

A drawer in a darkened room contains 100 red socks, 80 green socks, 60 blue socks and 40 black socks. A boy selects socks one at a time from the drawer but is unable to see the color of the socks drawn.

What is the smallest number of socks that must be selected to guarantee that the selection contains at least 10 matching pairs? ( No sock may be counted in more than one pair.)

2. Originally Posted by smoothi963
A drawer in a darkened room contains 100 red socks, 80 green socks, 60 blue socks and 40 black socks. A boy selects socks one at a time from the drawer but is unable to see the color of the socks drawn.

What is the smallest number of socks that must be selected to guarantee that the selection contains at least 10 matching pairs? ( No sock may be counted in more than one pair.)
As the first 4 socks could be r,g,bl,bk he must draw at least 5 socks to
ensure 1 pair (suppose this pair is bk). Now as the next sock may be bk
he must draw another 2 socks to ensure another pair (total 6 socks drawn)
and so on for each additional pair.

In general he will have to draw 5+2*(n-1) socks to guaranty getting n pairs

So to get 10 pairs he will need to draw 23 socks.

RonL

3. Hello, smoothi963!

Another approach . . .

A drawer in a darkened room contains 100 red socks, 80 green socks,
60 blue socks and 40 black socks.
A boy selects socks one at a time from the drawer
but is unable to see the color of the socks drawn.

What is the smallest number of socks that must be selected to guarantee
that the selection contains at least 10 matching pairs?
(No sock may be counted in more than one pair.)

Consider the worst-case scenario . . .

He has nine matching pairs (18 socks)
. . and one each of the four colors (red, green, blue, black).
A total of 22 socks and does not have ten matching pairs.

However, the next sock will give him ten matching pairs.

Therefore, he must draw 23 socks to get ten matching pairs.