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Combination Lock Possibilities
Name: Farhad D.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
Brain teaser question I gave to my students that has
actually got me confused. The question is... This type of
combination lock has six numerals on it. To open the lock requires
three moves-right, left, right. For example one combination is right
to 3, left to 0 and right to 5. How many different combinations are
possible? I worked it out as first step has 6 possibilities, second
step has 36 for each one numeral so 36 x 6 =216, and final step has
216 x 6 for a total of 1296, then I add all three steps 6 + 216 +
1296 for a total of 1518 possibilities. Am I correct?? If not could
you please explain the right way. Also if the lock had 60 numerals
would I simply multiply the answer by ten for a total of 12960?
Replies:
Farhad D.,
As I interpret this problem, imagine six buttons in a row, numbered zero
to five. The first button you push must be to the right of the zero so
that you can move left afterwards. The second button must be to the
left of the first button. The third button must be to the right of the
second.
Consider the options. Look for a pattern.
1,0, (choice of from 1 to 5) 5
2,0, (choice of from 1 to 5) 5
2,1, (choice of from 2 to 5) 4
3,0 5
3,1 4
3,2 3
Etc.
The number of options is
(5)+(5+4)+(5+4+3)+(5+4+3+2)+(5+4+3+2+1)=5*5+4*4+3*3+2*2+1*1=25+16+9+4+1=
55
IF you had more numbers, you would have the same pattern. 60 buttons (0
to 59) would yield:
59*59+58*58+57*57+...+3*3+2*2+1*1.
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
As I understand the question, to open the lock requires a right move,
a left move and the a right move. All three are required. If there are
6 choices for each move, the would be 6 times 6 times 6 or 216 total
possibilities. If there were 60 numbers to select from, there would be
60 times 60 times 60 ways or 216 times 1000 or 216,000 ways.
R. Paul Beem
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Update: June 2012
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