Difference between revisions of "1986 AHSME Problems/Problem 17"

m (Problem)
(Solution)
 
(3 intermediate revisions by 3 users not shown)
Line 13: Line 13:
  
 
==Solution==
 
==Solution==
 +
Solution by e_power_pi_times_i
  
 +
 +
Suppose that you wish to draw one pair of socks from the drawer. Then you would pick <math>5</math> socks (one of each kind, plus one). Notice that in the worst possible situation, you will continue to draw the same sock, until you get <math>10</math> pairs. This is because drawing the same sock results in a pair every <math>2</math> of that sock, whereas drawing another sock creates another pair. Thus the answer is <math>5+2\cdot(10-1) = \boxed{\textbf{(B) } 23}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 19:18, 20 July 2020

Problem

A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster 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 $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)

$\textbf{(A)}\ 21\qquad \textbf{(B)}\ 23\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ 50$

Solution

Solution by e_power_pi_times_i


Suppose that you wish to draw one pair of socks from the drawer. Then you would pick $5$ socks (one of each kind, plus one). Notice that in the worst possible situation, you will continue to draw the same sock, until you get $10$ pairs. This is because drawing the same sock results in a pair every $2$ of that sock, whereas drawing another sock creates another pair. Thus the answer is $5+2\cdot(10-1) = \boxed{\textbf{(B) } 23}$.

See also

1986 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS