Discrete Mathematics - Combinations/Factorials?

[aliaze1]

Homework Statement

An electronic switch bank consists of a row of six on - off switches. How many different
settings are possible if exactly three of the switches are set to off?

(a) 12 (b) 144 (c) 60 (d) 30 (e) 20

Homework Equations

Factorial rule?

5!=5*4*3*2*1

The Attempt at a Solution

There are 6 switches, but only two positions, so would it be 6*5=30?

 

[dynamicsolo]

Any one switch is either on or off, but in a problem like this, you would treat those as categories (like a coin landing heads or tails). The issue here is that only three of the six switches are off (also exactly three are on) and it matters which three those are.

Think of the six switches as slots which can be assigned either an 'on' or 'off' value. Take the first of the three 'offs': how many slots are there to assign that to? Once you've done that, how many remain to assign the second 'off' to? How about for the third? How many possible ways could you do this?

Now, once any particular assignment of the three 'offs' has been made, does it matter in any important way what order those 'offs' were assigned to each slot? If it doesn't, you need to divide your earlier answer to the question in the previous paragraph by the number of orders in which you could have assigned those three 'offs'. That will then answer your problem.

 

[Architeuthis Dux]

Your attempt at a solution is close, but not quite correct. The correct solution would be to use the combination formula, which takes into account the number of switches and the number of positions (on or off). The formula is nCr = n! / (r!*(n-r)!), where n is the total number of switches and r is the number of switches set to off. In this case, n=6 and r=3, so the solution would be 6C3 = 6! / (3!*(6-3)!) = (6*5*4) / (3*2*1) = 20. Therefore, the correct answer is (e) 20. This is an example of a combination, where order does not matter, as opposed to a permutation, where order does matter. In this case, we are choosing 3 switches out of a total of 6, so order does not matter and we use the combination formula.