How many times in a 12 hour period does the sum of the digits on a digital clock equal 6? try to think of a way to solve this without giving through every single time. describe what you did. plz answer this i will give you 100 points

Let the time be wx:yz (so at 11:51, w=1, x=1, y=5, z=1). We can describe the possible values for each variable:  

w: 0, 1  

x: 1, 2, 3, 4, 5, 6, 7, 8, 9 -- except, if w=1, then we know that x = 0, 1, 2  

y: 0, 1, 2, 3, 4, 5  

z: 0, 1, 2, 3, 4, 5, 6, 7, 8 , 9  

We're looking for all the possible w,x,y,z combinations that add up to 6.  

First, assume w=0, so it's between 1:00 and 9:59. Here we need x+y+z to be 6. We can immediately discard any numbers bigger than 6.  

So, write out all the combinations of 3 numbers that add up to 6:  

(6, 0, 0)  

(5, 1, 0 )  

(5, 0, 1)  

(4, 2, 0)  

(4, 0, 2)  

(4, 1, 1)  

(3, 3, 0)  

(3, 0, 3)  

(3, 2, 1)  

(3, 1, 2)  

(2, 4, 0)  

(2, 0, 4)  

(2, 3, 1)  

(2, 1, 3)  

(2, 2, 2)  

(1, 5, 0)  

(1, 0, 5)  

(1, 4, 1)  

(1, 1, 4)  

(1, 3, 2)  

(1, 2, 3)  

You might notice that for each value of x, there are 7-x times. That's because for each x, we need y+z to be 6-x, and there are n+1 ways to write n as the sum of two positive integers.  

Now, we need to think about the times when w=1. That means we need x+y+z to be 5, but remember that x can only be 0, 1, or 2.  

If x = 0, y+z=5, so there are 5+1 ways to do it (see if you can write them out.)  

if x=1, y+z = 4 so there are 4+1 ways to do it  

if x=2, y+z = 3 so there are 3+1 ways to do it.  

So, add up all those ways and we get 21 + 15 = 36 times.

europa2433

europa2433

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