Help please I tried solving this for a week...

Question:

There are several sets of three different numbers whose sum is 15 which can be chosen from {1,2,3,4,5,6,7,8,9}. How many of these sets contain a 5?

Edit: the answer is 4, I just want to know why...

https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_11

tommarvoloriddle Aug 29, 2019

#1**+2 **

I get four sets

{5 1 9}

{5 2 8}

{5 3 7}

{5 4 6}

{5, 5, 5} not possible

(5, 6, 4} repeat set

(5, 7, 3} repeat set

(5, 8, 2} repeat set

(5, 9 , 1} repeat set

CPhill Aug 30, 2019

#2**+5 **

These problems are really brute force, there is no real clever way. I only just listed all possible combinations

HOWEVER, notice in CPhill's answer.

5 19

5 2 8

5 3 7

5 4 6

Since ALL the sets contain a 5, we just subtract 5 from 15.

So we need TWO terms in a set and they add up to 10, and also are unique.

This is quite easy.

1 9

2 8

3 7

4 6

5 5 (doesnt work because cannot repeat digit)

CalculatorUser
Aug 30, 2019

#3**+1 **

Brute force!

Oh no I was hoping for a trick... :(

But your answers are still great!

tommarvoloriddle
Aug 30, 2019

#5**+4 **

I mean the trick is to find all unique pairs that added up to 10. But its KIND OF a trick.

CalculatorUser
Aug 30, 2019

#6**+3 **

I have a bad habit of not recognizing tricks, and usually brute force everything... Only when I review do I realize the clever tricks to problems. This is why I take so long on AMC competitions.

The worst kind of problems are the ones where you have to know a formula to solve, because there is no cleverness and you know it or you dont know it.

CalculatorUser
Aug 30, 2019

#7**0 **

I finished the AMC8.

But you know that pamphlet...

I put my answers there instead of in the bubble sheet...

XD

So I have to retake it...

tommarvoloriddle
Aug 30, 2019

#8**+2 **

OOOFF !!!

Thats a massive OOOOOFFFFF

I mean, AMC 8 is a very important test for me. And my parents and my teachers are worried on how I do this year, so seeing what happened to you made it seem scary.

CalculatorUser
Aug 30, 2019

#9**0 **

yeah but there is a problem-I skipped grades so its confusing...

Im in 8th grade now so im so confused.

still taking it

XD

tommarvoloriddle
Aug 30, 2019